arXiv subjects

Satyen Kale

Publications and source records attributed to Satyen Kale.

At least 37 records · Page 2Linked to original sources

arXiv2022

Pushing the Efficiency-Regret Pareto Frontier for Online Learning of Portfolios and Quantum States

We revisit the classical online portfolio selection problem. It is widely assumed that a trade-off between computational complexity and regret is unavoidable, with Cover's Universal Portfolios algorithm, SOFT-BAYES and ADA-BARRONS currently constituting its state-of-the-art Pareto frontier. In this paper, we present the first efficient algorithm, BISONS, that obtains polylogarithmic regret with memory and per-step running time requirements that are polynomial in the dimension, displacing ADA-BARRONS from the Pareto frontier. Additionally, we resolve a COLT 2020 open problem by showing that a certain Follow-The-Regularized-Leader algorithm with log-barrier regularization suffers an exponentially larger dependence on the dimension than previously conjectured. Thus, we rule out this algorithm as a candidate for the Pareto frontier. We also extend our algorithm and analysis to a more general problem than online portfolio selection, viz. online learning of quantum states with log loss. This algorithm, called SCHRODINGER'S BISONS, is the first efficient algorithm with polylogarithmic regret for this more general problem.

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arXiv2022

Agnostic Learnability of Halfspaces via Logistic Loss

We investigate approximation guarantees provided by logistic regression for the fundamental problem of agnostic learning of homogeneous halfspaces. Previously, for a certain broad class of "well-behaved" distributions on the examples, Diakonikolas et al. (2020) proved an $\tildeΩ(\textrm{OPT})$ lower bound, while Frei et al. (2021) proved an $\tilde{O}(\sqrt{\textrm{OPT}})$ upper bound, where $\textrm{OPT}$ denotes the best zero-one/misclassification risk of a homogeneous halfspace. In this paper, we close this gap by constructing a well-behaved distribution such that the global minimizer of the logistic risk over this distribution only achieves $Ω(\sqrt{\textrm{OPT}})$ misclassification risk, matching the upper bound in (Frei et al., 2021). On the other hand, we also show that if we impose a radial-Lipschitzness condition in addition to well-behaved-ness on the distribution, logistic regression on a ball of bounded radius reaches $\tilde{O}(\textrm{OPT})$ misclassification risk. Our techniques also show for any well-behaved distribution, regardless of radial Lipschitzness, we can overcome the $Ω(\sqrt{\textrm{OPT}})$ lower bound for logistic loss simply at the cost of one additional convex optimization step involving the hinge loss and attain $\tilde{O}(\textrm{OPT})$ misclassification risk. This two-step convex optimization algorithm is simpler than previous methods obtaining this guarantee, all of which require solving $O(\log(1/\textrm{OPT}))$ minimization problems.

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arXiv2021

Learning with User-Level Privacy

We propose and analyze algorithms to solve a range of learning tasks under user-level differential privacy constraints. Rather than guaranteeing only the privacy of individual samples, user-level DP protects a user's entire contribution ($m \ge 1$ samples), providing more stringent but more realistic protection against information leaks. We show that for high-dimensional mean estimation, empirical risk minimization with smooth losses, stochastic convex optimization, and learning hypothesis classes with finite metric entropy, the privacy cost decreases as $O(1/\sqrt{m})$ as users provide more samples. In contrast, when increasing the number of users $n$, the privacy cost decreases at a faster $O(1/n)$ rate. We complement these results with lower bounds showing the minimax optimality of our algorithms for mean estimation and stochastic convex optimization. Our algorithms rely on novel techniques for private mean estimation in arbitrary dimension with error scaling as the concentration radius $τ$ of the distribution rather than the entire range.

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arXiv2021

Efficient Methods for Online Multiclass Logistic Regression

Multiclass logistic regression is a fundamental task in machine learning with applications in classification and boosting. Previous work (Foster et al., 2018) has highlighted the importance of improper predictors for achieving "fast rates" in the online multiclass logistic regression problem without suffering exponentially from secondary problem parameters, such as the norm of the predictors in the comparison class. While Foster et al. (2018) introduced a statistically optimal algorithm, it is in practice computationally intractable due to its run-time complexity being a large polynomial in the time horizon and dimension of input feature vectors. In this paper, we develop a new algorithm, FOLKLORE, for the problem which runs significantly faster than the algorithm of Foster et al.(2018) -- the running time per iteration scales quadratically in the dimension -- at the cost of a linear dependence on the norm of the predictors in the regret bound. This yields the first practical algorithm for online multiclass logistic regression, resolving an open problem of Foster et al.(2018). Furthermore, we show that our algorithm can be applied to online bandit multiclass prediction and online multiclass boosting, yielding more practical algorithms for both problems compared to the ones in Foster et al.(2018) with similar performance guarantees. Finally, we also provide an online-to-batch conversion result for our algorithm.

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arXiv2021

A Field Guide to Federated Optimization

Federated learning and analytics are a distributed approach for collaboratively learning models (or statistics) from decentralized data, motivated by and designed for privacy protection. The distributed learning process can be formulated as solving federated optimization problems, which emphasize communication efficiency, data heterogeneity, compatibility with privacy and system requirements, and other constraints that are not primary considerations in other problem settings. This paper provides recommendations and guidelines on formulating, designing, evaluating and analyzing federated optimization algorithms through concrete examples and practical implementation, with a focus on conducting effective simulations to infer real-world performance. The goal of this work is not to survey the current literature, but to inspire researchers and practitioners to design federated learning algorithms that can be used in various practical applications.

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arXiv2021

SGD: The Role of Implicit Regularization, Batch-size and Multiple-epochs

Multi-epoch, small-batch, Stochastic Gradient Descent (SGD) has been the method of choice for learning with large over-parameterized models. A popular theory for explaining why SGD works well in practice is that the algorithm has an implicit regularization that biases its output towards a good solution. Perhaps the theoretically most well understood learning setting for SGD is that of Stochastic Convex Optimization (SCO), where it is well known that SGD learns at a rate of $O(1/\sqrt{n})$, where $n$ is the number of samples. In this paper, we consider the problem of SCO and explore the role of implicit regularization, batch size and multiple epochs for SGD. Our main contributions are threefold: (a) We show that for any regularizer, there is an SCO problem for which Regularized Empirical Risk Minimzation fails to learn. This automatically rules out any implicit regularization based explanation for the success of SGD. (b) We provide a separation between SGD and learning via Gradient Descent on empirical loss (GD) in terms of sample complexity. We show that there is an SCO problem such that GD with any step size and number of iterations can only learn at a suboptimal rate: at least $\widetildeΩ(1/n^{5/12})$. (c) We present a multi-epoch variant of SGD commonly used in practice. We prove that this algorithm is at least as good as single pass SGD in the worst case. However, for certain SCO problems, taking multiple passes over the dataset can significantly outperform single pass SGD. We extend our results to the general learning setting by showing a problem which is learnable for any data distribution, and for this problem, SGD is strictly better than RERM for any regularization function. We conclude by discussing the implications of our results for deep learning, and show a separation between SGD and ERM for two layer diagonal neural networks.

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arXiv2021

Mime: Mimicking Centralized Stochastic Algorithms in Federated Learning

Federated learning (FL) is a challenging setting for optimization due to the heterogeneity of the data across different clients which gives rise to the client drift phenomenon. In fact, obtaining an algorithm for FL which is uniformly better than simple centralized training has been a major open problem thus far. In this work, we propose a general algorithmic framework, Mime, which i) mitigates client drift and ii) adapts arbitrary centralized optimization algorithms such as momentum and Adam to the cross-device federated learning setting. Mime uses a combination of control-variates and server-level statistics (e.g. momentum) at every client-update step to ensure that each local update mimics that of the centralized method run on iid data. We prove a reduction result showing that Mime can translate the convergence of a generic algorithm in the centralized setting into convergence in the federated setting. Further, we show that when combined with momentum based variance reduction, Mime is provably faster than any centralized method--the first such result. We also perform a thorough experimental exploration of Mime's performance on real world datasets.

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arXiv2021

SCAFFOLD: Stochastic Controlled Averaging for Federated Learning

Federated Averaging (FedAvg) has emerged as the algorithm of choice for federated learning due to its simplicity and low communication cost. However, in spite of recent research efforts, its performance is not fully understood. We obtain tight convergence rates for FedAvg and prove that it suffers from `client-drift' when the data is heterogeneous (non-iid), resulting in unstable and slow convergence. As a solution, we propose a new algorithm (SCAFFOLD) which uses control variates (variance reduction) to correct for the `client-drift' in its local updates. We prove that SCAFFOLD requires significantly fewer communication rounds and is not affected by data heterogeneity or client sampling. Further, we show that (for quadratics) SCAFFOLD can take advantage of similarity in the client's data yielding even faster convergence. The latter is the first result to quantify the usefulness of local-steps in distributed optimization.

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arXiv2021

Federated Functional Gradient Boosting

In this paper, we initiate a study of functional minimization in Federated Learning. First, in the semi-heterogeneous setting, when the marginal distributions of the feature vectors on client machines are identical, we develop the federated functional gradient boosting (FFGB) method that provably converges to the global minimum. Subsequently, we extend our results to the fully-heterogeneous setting (where marginal distributions of feature vectors may differ) by designing an efficient variant of FFGB called FFGB.C, with provable convergence to a neighborhood of the global minimum within a radius that depends on the total variation distances between the client feature distributions. For the special case of square loss, but still in the fully heterogeneous setting, we design the FFGB.L method that also enjoys provable convergence to a neighborhood of the global minimum but within a radius depending on the much tighter Wasserstein-1 distances. For both FFGB.C and FFGB.L, the radii of convergence shrink to zero as the feature distributions become more homogeneous. Finally, we conduct proof-of-concept experiments to demonstrate the benefits of our approach against natural baselines.

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arXiv2021

A Multiclass Boosting Framework for Achieving Fast and Provable Adversarial Robustness

Alongside the well-publicized accomplishments of deep neural networks there has emerged an apparent bug in their success on tasks such as object recognition: with deep models trained using vanilla methods, input images can be slightly corrupted in order to modify output predictions, even when these corruptions are practically invisible. This apparent lack of robustness has led researchers to propose methods that can help to prevent an adversary from having such capabilities. The state-of-the-art approaches have incorporated the robustness requirement into the loss function, and the training process involves taking stochastic gradient descent steps not using original inputs but on adversarially-corrupted ones. In this paper we propose a multiclass boosting framework to ensure adversarial robustness. Boosting algorithms are generally well-suited for adversarial scenarios, as they were classically designed to satisfy a minimax guarantee. We provide a theoretical foundation for this methodology and describe conditions under which robustness can be achieved given a weak training oracle. We show empirically that adversarially-robust multiclass boosting not only outperforms the state-of-the-art methods, it does so at a fraction of the training time.

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arXiv2021

A Deep Conditioning Treatment of Neural Networks

We study the role of depth in training randomly initialized overparameterized neural networks. We give a general result showing that depth improves trainability of neural networks by improving the conditioning of certain kernel matrices of the input data. This result holds for arbitrary non-linear activation functions under a certain normalization. We provide versions of the result that hold for training just the top layer of the neural network, as well as for training all layers, via the neural tangent kernel. As applications of these general results, we provide a generalization of the results of Das et al. (2019) showing that learnability of deep random neural networks with a large class of non-linear activations degrades exponentially with depth. We also show how benign overfitting can occur in deep neural networks via the results of Bartlett et al. (2019b). We also give experimental evidence that normalized versions of ReLU are a viable alternative to more complex operations like Batch Normalization in training deep neural networks.

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arXiv2020

Estimating Training Data Influence by Tracing Gradient Descent

We introduce a method called TracIn that computes the influence of a training example on a prediction made by the model. The idea is to trace how the loss on the test point changes during the training process whenever the training example of interest was utilized. We provide a scalable implementation of TracIn via: (a) a first-order gradient approximation to the exact computation, (b) saved checkpoints of standard training procedures, and (c) cherry-picking layers of a deep neural network. In contrast with previously proposed methods, TracIn is simple to implement; all it needs is the ability to work with gradients, checkpoints, and loss functions. The method is general. It applies to any machine learning model trained using stochastic gradient descent or a variant of it, agnostic of architecture, domain and task. We expect the method to be widely useful within processes that study and improve training data.

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arXiv2020

Hypothesis Set Stability and Generalization

We present a study of generalization for data-dependent hypothesis sets. We give a general learning guarantee for data-dependent hypothesis sets based on a notion of transductive Rademacher complexity. Our main result is a generalization bound for data-dependent hypothesis sets expressed in terms of a notion of hypothesis set stability and a notion of Rademacher complexity for data-dependent hypothesis sets that we introduce. This bound admits as special cases both standard Rademacher complexity bounds and algorithm-dependent uniform stability bounds. We also illustrate the use of these learning bounds in the analysis of several scenarios.

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arXiv2020

Escaping Saddle Points with Adaptive Gradient Methods

Adaptive methods such as Adam and RMSProp are widely used in deep learning but are not well understood. In this paper, we seek a crisp, clean and precise characterization of their behavior in nonconvex settings. To this end, we first provide a novel view of adaptive methods as preconditioned SGD, where the preconditioner is estimated in an online manner. By studying the preconditioner on its own, we elucidate its purpose: it rescales the stochastic gradient noise to be isotropic near stationary points, which helps escape saddle points. Furthermore, we show that adaptive methods can efficiently estimate the aforementioned preconditioner. By gluing together these two components, we provide the first (to our knowledge) second-order convergence result for any adaptive method. The key insight from our analysis is that, compared to SGD, adaptive methods escape saddle points faster, and can converge faster overall to second-order stationary points.

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arXiv2019

Online Learning of Quantum States

Suppose we have many copies of an unknown $n$-qubit state $ρ$. We measure some copies of $ρ$ using a known two-outcome measurement $E_{1}$, then other copies using a measurement $E_{2}$, and so on. At each stage $t$, we generate a current hypothesis $σ_{t}$ about the state $ρ$, using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that $|\operatorname{Tr}(E_{i} σ_{t}) - \operatorname{Tr}(E_{i}ρ) |$, the error in our prediction for the next measurement, is at least $\varepsilon$ at most $\operatorname{O}\!\left(n / \varepsilon^2 \right) $ times. Even in the "non-realizable" setting---where there could be arbitrary noise in the measurement outcomes---we show how to output hypothesis states that do significantly worse than the best possible states at most $\operatorname{O}\!\left(\sqrt {Tn}\right) $ times on the first $T$ measurements. These results generalize a 2007 theorem by Aaronson on the PAC-learnability of quantum states, to the online and regret-minimization settings. We give three different ways to prove our results---using convex optimization, quantum postselection, and sequential fat-shattering dimension---which have different advantages in terms of parameters and portability.

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arXiv2019

On the Convergence of Adam and Beyond

Several recently proposed stochastic optimization methods that have been successfully used in training deep networks such as RMSProp, Adam, Adadelta, Nadam are based on using gradient updates scaled by square roots of exponential moving averages of squared past gradients. In many applications, e.g. learning with large output spaces, it has been empirically observed that these algorithms fail to converge to an optimal solution (or a critical point in nonconvex settings). We show that one cause for such failures is the exponential moving average used in the algorithms. We provide an explicit example of a simple convex optimization setting where Adam does not converge to the optimal solution, and describe the precise problems with the previous analysis of Adam algorithm. Our analysis suggests that the convergence issues can be fixed by endowing such algorithms with `long-term memory' of past gradients, and propose new variants of the Adam algorithm which not only fix the convergence issues but often also lead to improved empirical performance.

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arXiv2018

Logistic Regression: The Importance of Being Improper

Learning linear predictors with the logistic loss---both in stochastic and online settings---is a fundamental task in machine learning and statistics, with direct connections to classification and boosting. Existing "fast rates" for this setting exhibit exponential dependence on the predictor norm, and Hazan et al. (2014) showed that this is unfortunately unimprovable. Starting with the simple observation that the logistic loss is $1$-mixable, we design a new efficient improper learning algorithm for online logistic regression that circumvents the aforementioned lower bound with a regret bound exhibiting a doubly-exponential improvement in dependence on the predictor norm. This provides a positive resolution to a variant of the COLT 2012 open problem of McMahan and Streeter (2012) when improper learning is allowed. This improvement is obtained both in the online setting and, with some extra work, in the batch statistical setting with high probability. We also show that the improved dependence on predictor norm is near-optimal. Leveraging this improved dependency on the predictor norm yields the following applications: (a) we give algorithms for online bandit multiclass learning with the logistic loss with an $\tilde{O}(\sqrt{n})$ relative mistake bound across essentially all parameter ranges, thus providing a solution to the COLT 2009 open problem of Abernethy and Rakhlin (2009), and (b) we give an adaptive algorithm for online multiclass boosting with optimal sample complexity, thus partially resolving an open problem of Beygelzimer et al. (2015) and Jung et al. (2017). Finally, we give information-theoretic bounds on the optimal rates for improper logistic regression with general function classes, thereby characterizing the extent to which our improvement for linear classes extends to other parametric and even nonparametric settings.

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arXiv2018

Stochastic Negative Mining for Learning with Large Output Spaces

We consider the problem of retrieving the most relevant labels for a given input when the size of the output space is very large. Retrieval methods are modeled as set-valued classifiers which output a small set of classes for each input, and a mistake is made if the label is not in the output set. Despite its practical importance, a statistically principled, yet practical solution to this problem is largely missing. To this end, we first define a family of surrogate losses and show that they are calibrated and convex under certain conditions on the loss parameters and data distribution, thereby establishing a statistical and analytical basis for using these losses. Furthermore, we identify a particularly intuitive class of loss functions in the aforementioned family and show that they are amenable to practical implementation in the large output space setting (i.e. computation is possible without evaluating scores of all labels) by developing a technique called Stochastic Negative Mining. We also provide generalization error bounds for the losses in the family. Finally, we conduct experiments which demonstrate that Stochastic Negative Mining yields benefits over commonly used negative sampling approaches.

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