Collaborating Authors

Koren, Tomer

Private Stochastic Convex Optimization: Optimal Rates in Linear Time Machine Learning

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.

Robust Bi-Tempered Logistic Loss Based on Bregman Divergences

Neural Information Processing Systems

We introduce a temperature into the exponential function and replace the softmax output layer of the neural networks by a high-temperature generalization. Similarly, the logarithm in the 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 networks by our bi-temperature generalization of the 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 datasets.

Memory Efficient Adaptive Optimization

Neural Information Processing Systems

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.

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

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.

Disentangling Adaptive Gradient Methods from Learning Rates Machine Learning

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.

Prediction with Corrupted Expert Advice Machine Learning

Prediction with expert advice is perhaps the single most fundamental problem in online learning and sequential decision making. In this problem, the goal of a learner is to aggregate decisions from multiple experts and achieve performance that approaches that of the best individual expert in hindsight. The standard performance criterion is the regret: the difference between the loss of the learner and that of the best single expert. The experts problem is often considered in the so-called adversarial setting, where the losses of the individual experts may be virtually arbitrary and even be chosen by an adversary so as to maximize the learner's regret. The canonical algorithm in this setup is the Multiplicative Weights algorithm (Littlestone and Warmuth, 1989; Freund and Schapire, 1995), that guarantees an optimal regret of Θ( T log N) in any problem with N experts and T decision rounds. A long line of research in online learning has focused on obtaining better regret guarantees, often referred to as "fast rates," on benign problem instances in which the loss generation process behaves more favourably than in a fully adversarial setup. A prototypical example of such an instance is the stochastic setting of the experts problem, where the losses of the experts are drawn i.i.d.

Second Order Optimization Made Practical Machine Learning

Second-order gradient methods are among the most powerful algorithms in mathematical optimization. Algorithms in this family use a preconditioner matrix to transform the gradient before applying each step. Classically, this involves computing or approximating the matrix of second-order derivatives, i.e, the Hessian, in the context of exact deterministic optimization (e.g., Fletcher, 2013; Lewis & Overton, 2013; Nocedal, 1980). In contrast, AdaGrad (Duchi et al., 2011) and related algorithms that target stochastic optimization use the covariance matrix of second-order gradient statistics to form the preconditioner. While second-order methods often have significantly better convergence properties than first-order methods, the size of typical problems prohibits their use in practice, as they require quadratic storage and cubic computation time for each gradient update. Thus, these methods not commonly seen in the present practice of optimization in machine learning, which is largely dominated by the simpler to implement first-order methods. Arguably, one of the greatest challenges of modern optimization is to bridge this gap between the theoretical and practical optimization and make second-order optimization more feasible to implement and deploy. In this paper, we attempt to contribute towards narrowing this gap between theory and practice, focusing on second-order adaptive methods. These methods can be thought of as full-matrix analogues of common adaptive algorithms of the family of AdaGrad (Duchi et al., 2011) and Adam (Kingma & Ba, 2014).

Logarithmic Regret for Learning Linear Quadratic Regulators Efficiently Machine Learning

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.

Beating SGD: Learning SVMs in Sublinear Time

Neural Information Processing Systems

We present an optimization approach for linear SVMs based on a stochastic primal-dual approach, where the primal step is akin to an importance-weighted SGD, and the dual step is a stochastic update on the importance weights. This yields an optimization method with a sublinear dependence on the training set size, and the first method for learning linear SVMs with runtime less then the size of the training set required for learning! Papers published at the Neural Information Processing Systems Conference.

Affine-Invariant Online Optimization and the Low-rank Experts Problem

Neural Information Processing Systems

We present a new affine-invariant optimization algorithm called Online Lazy Newton. The regret of Online Lazy Newton is independent of conditioning: the algorithm's performance depends on the best possible preconditioning of the problem in retrospect and on its \emph{intrinsic} dimensionality. As an application, we show how Online Lazy Newton can be used to achieve an optimal regret of order $\sqrt{rT}$ for the low-rank experts problem, improving by a $\sqrt{r}$ factor over the previously best known bound and resolving an open problem posed by Hazan et al (2016). Papers published at the Neural Information Processing Systems Conference.