We provide novel theoretical insights on structured prediction in the context of efficient convex surrogate loss minimization with consistency guarantees. For any task loss, we construct a convex surrogate that can be optimized via stochastic gradient descent and we prove tight bounds on the so-called "calibration function" relating the excess surrogate risk to the actual risk. In contrast to prior related work, we carefully monitor the effect of the exponential number of classes in the learning guarantees as well as on the optimization complexity. As an interesting consequence, we formalize the intuition that some task losses make learning harder than others, and that the classical 0-1 loss is ill-suited for structured prediction. Papers published at the Neural Information Processing Systems Conference.

In deterministic optimization, line searches are a standard tool ensuring stability and efficiency. Where only stochastic gradients are available, no direct equivalent has so far been formulated, because uncertain gradients do not allow for a strict sequence of decisions collapsing the search space. We construct a probabilistic line search by combining the structure of existing deterministic methods with notions from Bayesian optimization. Our method retains a Gaussian process surrogate of the univariate optimization objective, and uses a probabilistic belief over the Wolfe conditions to monitor the descent. The algorithm has very low computational cost, and no user-controlled parameters.

We propose a simple, scalable, and fast gradient descent algorithm to optimize a nonconvex objective for the rank minimization problem and a closely related family of semidefinite programs. With $O(r 3 \kappa 2 n \log n)$ random measurements of a positive semidefinite $n\times n$ matrix of rank $r$ and condition number $\kappa$, our method is guaranteed to converge linearly to the global optimum. Papers published at the Neural Information Processing Systems Conference.

We study the resilience to Byzantine failures of distributed implementations of Stochastic Gradient Descent (SGD). So far, distributed machine learning frameworks have largely ignored the possibility of failures, especially arbitrary (i.e., Byzantine) ones. Causes of failures include software bugs, network asynchrony, biases in local datasets, as well as attackers trying to compromise the entire system. Assuming a set of $n$ workers, up to $f$ being Byzantine, we ask how resilient can SGD be, without limiting the dimension, nor the size of the parameter space. We first show that no gradient aggregation rule based on a linear combination of the vectors proposed by the workers (i.e, current approaches) tolerates a single Byzantine failure.

Stochastic gradient methods for machine learning and optimization problems are usually analyzed assuming data points are sampled *with* replacement. In contrast, sampling *without* replacement is far less understood, yet in practice it is very common, often easier to implement, and usually performs better. In this paper, we provide competitive convergence guarantees for without-replacement sampling under several scenarios, focusing on the natural regime of few passes over the data. Moreover, we describe a useful application of these results in the context of distributed optimization with randomly-partitioned data, yielding a nearly-optimal algorithm for regularized least squares (in terms of both communication complexity and runtime complexity) under broad parameter regimes. Our proof techniques combine ideas from stochastic optimization, adversarial online learning and transductive learning theory, and can potentially be applied to other stochastic optimization and learning problems. Papers published at the Neural Information Processing Systems Conference.

Due to their simplicity and excellent performance, parallel asynchronous variants of stochastic gradient descent have become popular methods to solve a wide range of large-scale optimization problems on multi-core architectures. Yet, despite their practical success, support for nonsmooth objectives is still lacking, making them unsuitable for many problems of interest in machine learning, such as the Lasso, group Lasso or empirical risk minimization with convex constraints. In this work, we propose and analyze ProxASAGA, a fully asynchronous sparse method inspired by SAGA, a variance reduced incremental gradient algorithm. The proposed method is easy to implement and significantly outperforms the state of the art on several nonsmooth, large-scale problems. We prove that our method achieves a theoretical linear speedup with respect to the sequential version under assumptions on the sparsity of gradients and block-separability of the proximal term.

Training a classifier over a large number of classes, known as 'extreme classification', has become a topic of major interest with applications in technology, science, and e-commerce. Traditional softmax regression induces a gradient cost proportional to the number of classes $C$, which often is prohibitively expensive. A popular scalable softmax approximation relies on uniform negative sampling, which suffers from slow convergence due a poor signal-to-noise ratio. In this paper, we propose a simple training method for drastically enhancing the gradient signal by drawing negative samples from an adversarial model that mimics the data distribution. Our contributions are three-fold: (i) an adversarial sampling mechanism that produces negative samples at a cost only logarithmic in $C$, thus still resulting in cheap gradient updates; (ii) a mathematical proof that this adversarial sampling minimizes the gradient variance while any bias due to non-uniform sampling can be removed; (iii) experimental results on large scale data sets that show a reduction of the training time by an order of magnitude relative to several competitive baselines.

We introduce a simple (one line of code) modification to the Generative Adversarial Network (GAN) training algorithm that materially improves results with no increase in computational cost: When updating the generator parameters, we simply zero out the gradient contributions from the elements of the batch that the critic scores as `least realistic'. Through experiments on many different GAN variants, we show that this `top-k update' procedure is a generally applicable improvement. In order to understand the nature of the improvement, we conduct extensive analysis on a simple mixture-of-Gaussians dataset and discover several interesting phenomena. Among these is that, when gradient updates are computed using the worst-scoring batch elements, samples can actually be pushed further away from the their nearest mode.

Optimization is a central ingredient of machine learning. First-order optimization algorithms, for instance, are particularly popular for deep learning tasks due to their scalabilities to highdimensional problems, because they employ gradient but not higher-order information of objective functions for iteratively approximating minimizers. Among first-order methods, arguably the most used is gradient descent method (GD), or rather one of its variants, stochastic gradient descent method (SGD). Designed for objective functions that sum a large amount of terms, which for instance can originate from big data, SGD introduces a randomization mechanism of gradient subsampling to improve the scalability of GD (e.g., Zhang [2004], Moulines and Bach [2011], Roux et al. [2012]). Consequently, the iteration of SGD, unlike GD, is not deterministic even when it is started at a fixed initial condition.

We introduce a sampling perspective to tackle the challenging task of training robust Reinforcement Learning (RL) agents. Leveraging the powerful Stochastic Gradient Langevin Dynamics, we present a novel, scalable two-player RL algorithm, which is a sampling variant of the two-player policy gradient method. Our algorithm consistently outperforms existing baselines, in terms of generalization across different training and testing conditions, on several MuJoCo environments. Our experiments also show that, even for objective functions that entirely ignore potential environmental shifts, our sampling approach remains highly robust in comparison to standard RL algorithms.