In this paper, we introduce a discrete variant of the meta-learning framework. Meta-learning aims at exploiting prior experience and data to improve performance on future tasks. By now, there exist numerous formulations for meta-learning in the continuous domain. Notably, the Model-Agnostic Meta-Learning (MAML) formulation views each task as a continuous optimization problem and based on prior data learns a suitable initialization that can be adapted to new, unseen tasks after a few simple gradient updates. Motivated by this terminology, we propose a novel meta-learning framework in the discrete domain where each task is equivalent to maximizing a set function under a cardinality constraint. Our approach aims at using prior data, i.e., previously visited tasks, to train a proper initial solution set that can be quickly adapted to a new task at a relatively low computational cost. This approach leads to (i) a personalized solution for each individual task, and (ii) significantly reduced computational cost at test time compared to the case where the solution is fully optimized once the new task is revealed. The training procedure is performed by solving a challenging discrete optimization problem for which we present deterministic and randomized algorithms. In the case where the tasks are monotone and submodular, we show strong theoretical guarantees for our proposed methods even though the training objective may not be submodular. We also demonstrate the effectiveness of our framework on two real-world problem instances where we observe that our methods lead to a significant reduction in computational complexity in solving the new tasks while incurring a small performance loss compared to when the tasks are fully optimized.

The existence of noisy labels in the dataset causes significant performance degradation for deep neural networks (DNNs). To address this problem, we propose a Meta Soft Label Generation algorithm called MSLG, which can jointly generate soft labels using meta-learning techniques and learn DNN parameters in an end-to-end fashion. Our approach adapts the meta-learning paradigm to estimate optimal label distribution by checking gradient directions on both noisy training data and noise-free meta-data. In order to iteratively update soft labels, meta-gradient descent step is performed on estimated labels, which would minimize the loss of noise-free meta samples. In each iteration, the base classifier is trained on estimated meta labels. MSLG is model-agnostic and can be added on top of any existing model at hand with ease. We performed extensive experiments on CIFAR10, Clothing1M and Food101N datasets. Results show that our approach outperforms other state-of-the-art methods by a large margin.

Reinforcement learning with function approximation can be unstable and even divergent, especially when combined with off-policy learning and Bellman updates. In deep reinforcement learning, these issues have been dealt with empirically by adapting and regularizing the representation, in particular with auxiliary tasks. This suggests that representation learning may provide a means to guarantee stability. In this paper, we formally show that there are indeed nontrivial state representations under which the canonical TD algorithm is stable, even when learning off-policy. We analyze representation learning schemes that are based on the transition matrix of a policy, such as proto-value functions, along three axes: approximation error, stability, and ease of estimation. In the most general case, we show that a Schur basis provides convergence guarantees, but is difficult to estimate from samples. For a fixed reward function, we find that an orthogonal basis of the corresponding Krylov subspace is an even better choice. We conclude by empirically demonstrating that these stable representations can be learned using stochastic gradient descent, opening the door to improved techniques for representation learning with deep networks.

When training the parameters of a linear dynamical model, the gradient descent algorithm is likely to fail to converge if the squared-error loss is used as the training loss function. Restricting the parameter space to a smaller subset and running the gradient descent algorithm within this subset can allow learning stable dynamical systems, but this strategy does not work for unstable systems. In this work, we look into the dynamics of the gradient descent algorithm and pinpoint what causes the difficulty of learning unstable systems. We show that observations taken at different times from the system to be learned influence the dynamics of the gradient descent algorithm in substantially different degrees. We introduce a time-weighted logarithmic loss function to fix this imbalance and demonstrate its effectiveness in learning unstable systems.

Neural network training is commonly accelerated by using multiple synchronized workers to compute gradient updates in parallel. Asynchronous methods remove synchronization overheads and improve hardware utilization at the cost of introducing gradient delay, which impedes optimization and can lead to lower final model performance. We introduce Adaptive Braking (AB), a modification for momentum-based optimizers that mitigates the effects of gradient delay. AB dynamically scales the gradient based on the alignment of the gradient and the velocity. This can dampen oscillations along high curvature directions of the loss surface, stabilizing and accelerating asynchronous training. We show that applying AB on top of SGD with momentum enables training ResNets on CIFAR-10 and ImageNet-1k with delays $D \geq$ 32 update steps with minimal drop in final test accuracy.

In this paper we tackle the challenge of making the stochastic coordinate descent algorithm differentially private. Compared to the classical gradient descent algorithm where updates operate on a single model vector and controlled noise addition to this vector suffices to hide critical information about individuals, stochastic coordinate descent crucially relies on keeping auxiliary information in memory during training. This auxiliary information provides an additional privacy leak and poses the major challenge addressed in this work. Driven by the insight that under independent noise addition, the consistency of the auxiliary information holds in expectation, we present DP-SCD, the first differentially private stochastic coordinate descent algorithm. We analyze our new method theoretically and argue that decoupling and parallelizing coordinate updates is essential for its utility. On the empirical side we demonstrate competitive performance against the popular stochastic gradient descent alternative (DP-SGD) while requiring significantly less tuning.

When using large-batch training to speed up stochastic gradient descent, learning rates must adapt to new batch sizes in order to maximize speed-ups and preserve model quality. Re-tuning learning rates is resource intensive, while fixed scaling rules often degrade model quality. We propose AdaScale SGD, an algorithm that reliably adapts learning rates to large-batch training. By continually adapting to the gradient's variance, AdaScale automatically achieves speed-ups for a wide range of batch sizes. We formally describe this quality with AdaScale's convergence bound, which maintains final objective values, even as batch sizes grow large and the number of iterations decreases. In empirical comparisons, AdaScale trains well beyond the batch size limits of popular "linear learning rate scaling" rules. This includes large-batch training with no model degradation for machine translation, image classification, object detection, and speech recognition tasks. AdaScale's qualitative behavior is similar to that of "warm-up" heuristics, but unlike warm-up, this behavior emerges naturally from a principled mechanism. The algorithm introduces negligible computational overhead and no new hyperparameters, making AdaScale an attractive choice for large-scale training in practice.

Intensive Longitudinal Data (ILD) is an increasingly common data type in the social and behavioral sciences. Despite the many benefits these data provide, little work has been dedicated to realizing the potential such data hold for forecasting dynamic processes at the individual level. To address this gap in the literature we present the multi-VAR framework, a novel methodological approach for penalized estimation and forecasting of ILD collected from multiple individuals. Importantly, our approach estimates models for all individuals simultaneously and is capable of adaptively adjusting to the amount of heterogeneity exhibited across individual dynamic processes. To accomplish this we propose proximal gradient descent algorithm for solving the multi-VAR problem and prove the consistency of the recovered transition matrices. We evaluate the forecasting performance of our method in comparison with a number of benchmark forecasting methods and provide an illustrative example involving the day-to-day emotional experiences of 16 individuals over an 11-week period.

We consider the dynamic of gradient descent for learning a two-layer neural network. We assume the input $x\in\mathbb{R}^d$ is drawn from a Gaussian distribution and the label of $x$ satisfies $f^{\star}(x) = a^{\top}|W^{\star}x|$, where $a\in\mathbb{R}^d$ is a nonnegative vector and $W^{\star} \in\mathbb{R}^{d\times d}$ is an orthonormal matrix. We show that an over-parametrized two-layer neural network with ReLU activation, trained by gradient descent from random initialization, can provably learn the ground truth network with population loss at most $o(1/d)$ in polynomial time with polynomial samples. On the other hand, we prove that any kernel method, including Neural Tangent Kernel, with a polynomial number of samples in $d$, has population loss at least $\Omega(1 / d)$.

The success of adversarial formulations in machine learning has brought renewed motivation for smooth games. In this work, we focus on the class of stochastic Hamiltonian methods and provide the first convergence guarantees for certain classes of stochastic smooth games. We propose a novel unbiased estimator for the stochastic Hamiltonian gradient descent (SHGD) and highlight its benefits. Using tools from the optimization literature we show that SHGD converges linearly to the neighbourhood of a stationary point. To guarantee convergence to the exact solution, we analyze SHGD with a decreasing step-size and we also present the first stochastic variance reduced Hamiltonian method. Our results provide the first global non-asymptotic last-iterate convergence guarantees for the class of stochastic unconstrained bilinear games and for the more general class of stochastic games that satisfy a "sufficiently bilinear" condition, notably including some non-convex non-concave problems. We supplement our analysis with experiments on stochastic bilinear and sufficiently bilinear games, where our theory is shown to be tight, and on simple adversarial machine learning formulations.