arXiv subjects

Tomer Koren

Publications and source records attributed to Tomer Koren.

At least 73 records · Page 4Linked to original sources

arXiv2021

Stochastic Multi-Armed Bandits with Unrestricted Delay Distributions

We study the stochastic Multi-Armed Bandit (MAB) problem with random delays in the feedback received by the algorithm. We consider two settings: the reward-dependent delay setting, where realized delays may depend on the stochastic rewards, and the reward-independent delay setting. Our main contribution is algorithms that achieve near-optimal regret in each of the settings, with an additional additive dependence on the quantiles of the delay distribution. Our results do not make any assumptions on the delay distributions: in particular, we do not assume they come from any parametric family of distributions and allow for unbounded support and expectation; we further allow for infinite delays where the algorithm might occasionally not observe any feedback.

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arXiv2021

Scalable Second Order Optimization for Deep Learning

Optimization in machine learning, both theoretical and applied, is presently dominated by first-order gradient methods such as stochastic gradient descent. Second-order optimization methods, that involve second derivatives and/or second order statistics of the data, are far less prevalent despite strong theoretical properties, due to their prohibitive computation, memory and communication costs. In an attempt to bridge this gap between theoretical and practical optimization, we present a scalable implementation of a second-order preconditioned method (concretely, a variant of full-matrix Adagrad), that along with several critical algorithmic and numerical improvements, provides significant convergence and wall-clock time improvements compared to conventional first-order methods on state-of-the-art deep models. Our novel design effectively utilizes the prevalent heterogeneous hardware architecture for training deep models, consisting of a multicore CPU coupled with multiple accelerator units. We demonstrate superior performance compared to state-of-the-art on very large learning tasks such as machine translation with Transformers, language modeling with BERT, click-through rate prediction on Criteo, and image classification on ImageNet with ResNet-50.

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arXiv2021

Private Stochastic Convex Optimization: Optimal Rates in $\ell_1$ Geometry

Stochastic convex optimization over an $\ell_1$-bounded domain is ubiquitous in machine learning applications such as LASSO but remains poorly understood when learning with differential privacy. We show that, up to logarithmic factors the optimal excess population loss of any $(\varepsilon,δ)$-differentially private optimizer is $\sqrt{\log(d)/n} + \sqrt{d}/\varepsilon n.$ The upper bound is based on a new algorithm that combines the iterative localization approach of~\citet{FeldmanKoTa20} with a new analysis of private regularized mirror descent. It applies to $\ell_p$ bounded domains for $p\in [1,2]$ and queries at most $n^{3/2}$ gradients improving over the best previously known algorithm for the $\ell_2$ case which needs $n^2$ gradients. Further, we show that when the loss functions satisfy additional smoothness assumptions, the excess loss is upper bounded (up to logarithmic factors) by $\sqrt{\log(d)/n} + (\log(d)/\varepsilon n)^{2/3}.$ This bound is achieved by a new variance-reduced version of the Frank-Wolfe algorithm that requires just a single pass over the data. We also show that the lower bound in this case is the minimum of the two rates mentioned above.

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arXiv2021

Online Policy Gradient for Model Free Learning of Linear Quadratic Regulators with $\sqrt{T}$ Regret

We consider the task of learning to control a linear dynamical system under fixed quadratic costs, known as the Linear Quadratic Regulator (LQR) problem. While model-free approaches are often favorable in practice, thus far only model-based methods, which rely on costly system identification, have been shown to achieve regret that scales with the optimal dependence on the time horizon T. We present the first model-free algorithm that achieves similar regret guarantees. Our method relies on an efficient policy gradient scheme, and a novel and tighter analysis of the cost of exploration in policy space in this setting.

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arXiv2021

Online Markov Decision Processes with Aggregate Bandit Feedback

We study a novel variant of online finite-horizon Markov Decision Processes with adversarially changing loss functions and initially unknown dynamics. In each episode, the learner suffers the loss accumulated along the trajectory realized by the policy chosen for the episode, and observes aggregate bandit feedback: the trajectory is revealed along with the cumulative loss suffered, rather than the individual losses encountered along the trajectory. Our main result is a computationally efficient algorithm with $O(\sqrt{K})$ regret for this setting, where $K$ is the number of episodes. We establish this result via an efficient reduction to a novel bandit learning setting we call Distorted Linear Bandits (DLB), which is a variant of bandit linear optimization where actions chosen by the learner are adversarially distorted before they are committed. We then develop a computationally-efficient online algorithm for DLB for which we prove an $O(\sqrt{T})$ regret bound, where $T$ is the number of time steps. Our algorithm is based on online mirror descent with a self-concordant barrier regularization that employs a novel increasing learning rate schedule.

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arXiv2020

Can Implicit Bias Explain Generalization? Stochastic Convex Optimization as a Case Study

The notion of implicit bias, or implicit regularization, has been suggested as a means to explain the surprising generalization ability of modern-days overparameterized learning algorithms. This notion refers to the tendency of the optimization algorithm towards a certain structured solution that often generalizes well. Recently, several papers have studied implicit regularization and were able to identify this phenomenon in various scenarios. We revisit this paradigm in arguably the simplest non-trivial setup, and study the implicit bias of Stochastic Gradient Descent (SGD) in the context of Stochastic Convex Optimization. As a first step, we provide a simple construction that rules out the existence of a \emph{distribution-independent} implicit regularizer that governs the generalization ability of SGD. We then demonstrate a learning problem that rules out a very general class of \emph{distribution-dependent} implicit regularizers from explaining generalization, which includes strongly convex regularizers as well as non-degenerate norm-based regularizations. Certain aspects of our constructions point out to significant difficulties in providing a comprehensive explanation of an algorithm's generalization performance by solely arguing about its implicit regularization properties.

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arXiv2020

Adversarial Dueling Bandits

We introduce the problem of regret minimization in Adversarial Dueling Bandits. As in classic Dueling Bandits, the learner has to repeatedly choose a pair of items and observe only a relative binary `win-loss' feedback for this pair, but here this feedback is generated from an arbitrary preference matrix, possibly chosen adversarially. Our main result is an algorithm whose $T$-round regret compared to the \emph{Borda-winner} from a set of $K$ items is $\tilde{O}(K^{1/3}T^{2/3})$, as well as a matching $Ω(K^{1/3}T^{2/3})$ lower bound. We also prove a similar high probability regret bound. We further consider a simpler \emph{fixed-gap} adversarial setup, which bridges between two extreme preference feedback models for dueling bandits: stationary preferences and an arbitrary sequence of preferences. For the fixed-gap adversarial setup we give an $\smash{ \tilde{O}((K/Δ^2)\log{T}) }$ regret algorithm, where $Δ$ is the gap in Borda scores between the best item and all other items, and show a lower bound of $Ω(K/Δ^2)$ indicating that our dependence on the main problem parameters $K$ and $Δ$ is tight (up to logarithmic factors).

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arXiv2020

Stochastic Optimization with Laggard Data Pipelines

State-of-the-art optimization is steadily shifting towards massively parallel pipelines with extremely large batch sizes. As a consequence, CPU-bound preprocessing and disk/memory/network operations have emerged as new performance bottlenecks, as opposed to hardware-accelerated gradient computations. In this regime, a recently proposed approach is data echoing (Choi et al., 2019), which takes repeated gradient steps on the same batch while waiting for fresh data to arrive from upstream. We provide the first convergence analyses of "data-echoed" extensions of common optimization methods, showing that they exhibit provable improvements over their synchronous counterparts. Specifically, we show that in convex optimization with stochastic minibatches, data echoing affords speedups on the curvature-dominated part of the convergence rate, while maintaining the optimal statistical rate.

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arXiv2020

Prediction with Corrupted Expert Advice

We revisit the fundamental problem of prediction with expert advice, in a setting where the environment is benign and generates losses stochastically, but the feedback observed by the learner is subject to a moderate adversarial corruption. We prove that a variant of the classical Multiplicative Weights algorithm with decreasing step sizes achieves constant regret in this setting and performs optimally in a wide range of environments, regardless of the magnitude of the injected corruption. Our results reveal a surprising disparity between the often comparable Follow the Regularized Leader (FTRL) and Online Mirror Descent (OMD) frameworks: we show that for experts in the corrupted stochastic regime, the regret performance of OMD is in fact strictly inferior to that of FTRL.

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arXiv2020

Holdout SGD: Byzantine Tolerant Federated Learning

This work presents a new distributed Byzantine tolerant federated learning algorithm, HoldOut SGD, for Stochastic Gradient Descent (SGD) optimization. HoldOut SGD uses the well known machine learning technique of holdout estimation, in a distributed fashion, in order to select parameter updates that are likely to lead to models with low loss values. This makes it more effective at discarding Byzantine workers inputs than existing methods that eliminate outliers in the parameter-space of the learned model. HoldOut SGD first randomly selects a set of workers that use their private data in order to propose gradient updates. Next, a voting committee of workers is randomly selected, and each voter uses its private data as holdout data, in order to select the best proposals via a voting scheme. We propose two possible mechanisms for the coordination of workers in the distributed computation of HoldOut SGD. The first uses a truthful central server and corresponds to the typical setting of current federated learning. The second is fully distributed and requires no central server, paving the way to fully decentralized federated learning. The fully distributed version implements HoldOut SGD via ideas from the blockchain domain, and specifically the Algorand committee selection and consensus processes. We provide formal guarantees for the HoldOut SGD process in terms of its convergence to the optimal model, and its level of resilience to the fraction of Byzantine workers. Empirical evaluation shows that HoldOut SGD is Byzantine-resilient and efficiently converges to an effectual model for deep-learning tasks, as long as the total number of participating workers is large and the fraction of Byzantine workers is less than half (<1/3 for the fully distributed variant).

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arXiv2020

Logarithmic Regret for Learning Linear Quadratic Regulators Efficiently

We consider the problem of learning in Linear Quadratic Control systems whose transition parameters are initially unknown. Recent results in this setting have demonstrated efficient learning algorithms with regret growing with the square root of the number of decision steps. We present new efficient algorithms that achieve, perhaps surprisingly, regret that scales only (poly)logarithmically with the number of steps in two scenarios: when only the state transition matrix $A$ is unknown, and when only the state-action transition matrix $B$ is unknown and the optimal policy satisfies a certain non-degeneracy condition. On the other hand, we give a lower bound that shows that when the latter condition is violated, square root regret is unavoidable.

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arXiv2020

Bandit Linear Control

We consider the problem of controlling a known linear dynamical system under stochastic noise, adversarially chosen costs, and bandit feedback. Unlike the full feedback setting where the entire cost function is revealed after each decision, here only the cost incurred by the learner is observed. We present a new and efficient algorithm that, for strongly convex and smooth costs, obtains regret that grows with the square root of the time horizon $T$. We also give extensions of this result to general convex, possibly non-smooth costs, and to non-stochastic system noise. A key component of our algorithm is a new technique for addressing bandit optimization of loss functions with memory.

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arXiv2020

Private Stochastic Convex Optimization: Optimal Rates in Linear Time

We study differentially private (DP) algorithms for stochastic convex optimization: the problem of minimizing the population loss given i.i.d. samples from a distribution over convex loss functions. A recent work of Bassily et al. (2019) has established the optimal bound on the excess population loss achievable given $n$ samples. Unfortunately, their algorithm achieving this bound is relatively inefficient: it requires $O(\min\{n^{3/2}, n^{5/2}/d\})$ gradient computations, where $d$ is the dimension of the optimization problem. We describe two new techniques for deriving DP convex optimization algorithms both achieving the optimal bound on excess loss and using $O(\min\{n, n^2/d\})$ gradient computations. In particular, the algorithms match the running time of the optimal non-private algorithms. The first approach relies on the use of variable batch sizes and is analyzed using the privacy amplification by iteration technique of Feldman et al. (2018). The second approach is based on a general reduction to the problem of localizing an approximately optimal solution with differential privacy. Such localization, in turn, can be achieved using existing (non-private) uniformly stable optimization algorithms. As in the earlier work, our algorithms require a mild smoothness assumption. We also give a linear-time algorithm achieving the optimal bound on the excess loss for the strongly convex case, as well as a faster algorithm for the non-smooth case.

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arXiv2020

Disentangling Adaptive Gradient Methods from Learning Rates

We investigate several confounding factors in the evaluation of optimization algorithms for deep learning. Primarily, we take a deeper look at how adaptive gradient methods interact with the learning rate schedule, a notoriously difficult-to-tune hyperparameter which has dramatic effects on the convergence and generalization of neural network training. We introduce a "grafting" experiment which decouples an update's magnitude from its direction, finding that many existing beliefs in the literature may have arisen from insufficient isolation of the implicit schedule of step sizes. Alongside this contribution, we present some empirical and theoretical retrospectives on the generalization of adaptive gradient methods, aimed at bringing more clarity to this space.

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arXiv2019

Robust Bi-Tempered Logistic Loss Based on Bregman Divergences

We introduce a temperature into the exponential function and replace the softmax output layer of neural nets by a high temperature generalization. Similarly, the logarithm in the log loss we use for training is replaced by a low temperature logarithm. By tuning the two temperatures we create loss functions that are non-convex already in the single layer case. When replacing the last layer of the neural nets by our bi-temperature generalization of logistic loss, the training becomes more robust to noise. We visualize the effect of tuning the two temperatures in a simple setting and show the efficacy of our method on large data sets. Our methodology is based on Bregman divergences and is superior to a related two-temperature method using the Tsallis divergence.

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arXiv2019

Memory-Efficient Adaptive Optimization

Adaptive gradient-based optimizers such as Adagrad and Adam are crucial for achieving state-of-the-art performance in machine translation and language modeling. However, these methods maintain second-order statistics for each parameter, thus introducing significant memory overheads that restrict the size of the model being used as well as the number of examples in a mini-batch. We describe an effective and flexible adaptive optimization method with greatly reduced memory overhead. Our method retains the benefits of per-parameter adaptivity while allowing significantly larger models and batch sizes. We give convergence guarantees for our method, and demonstrate its effectiveness in training very large translation and language models with up to 2-fold speedups compared to the state-of-the-art.

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arXiv2019

Semi-Cyclic Stochastic Gradient Descent

We consider convex SGD updates with a block-cyclic structure, i.e. where each cycle consists of a small number of blocks, each with many samples from a possibly different, block-specific, distribution. This situation arises, e.g., in Federated Learning where the mobile devices available for updates at different times during the day have different characteristics. We show that such block-cyclic structure can significantly deteriorate the performance of SGD, but propose a simple approach that allows prediction with the same performance guarantees as for i.i.d., non-cyclic, sampling.

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arXiv2019

Better Algorithms for Stochastic Bandits with Adversarial Corruptions

We study the stochastic multi-armed bandits problem in the presence of adversarial corruption. We present a new algorithm for this problem whose regret is nearly optimal, substantially improving upon previous work. Our algorithm is agnostic to the level of adversarial contamination and can tolerate a significant amount of corruption with virtually no degradation in performance.

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