Yura Malitsky Konstantin Mishchenko † Abstract We present a strikingly simple proof that two rules are sufficient to automate gradient descent: 1) don't increase the stepsize too fast and 2) don't overstep the local curvature. By following these rules, you get a method adaptive to the local geometry, with convergence guarantees depending only on smoothness in a neighborhood of a solution. Given that the problem is convex, our method will converge even if the global smoothness constant is infinity. As an illustration, it can minimize arbitrary continuously twice-differentiable convex function. We examine its performance on a range of convex and nonconvex problems, including matrix factorization and training of ResNet-18. 1 Introduction Since the early days of optimization it was evident that there is a need for algorithms that are as independent from the user as possible. First-order methods have proven to be versatile and efficient in a wide range of applications, but one drawback has been present all that time: the stepsize. Despite some certain success stories, line search procedures and adaptive online methods have not removed the need to manually tune the optimization parameters. Even in smooth convex optimization, which is often believed to be much simpler than the nonconvex counterpart, robust rules for stepsize selection have been elusive. The purpose of this work is to remedy this deficiency. The problem formulation that we consider is the basic unconstrained optimization problem min x R d f (x), (1) where f: R d R is a differentiable function. Throughout the paper we assume that (1) has a solution and we denote its optimal value by f . The simplest and most known approach to this problem is the gradient descent method (GD), whose origin can be traced back to Cauchy [7,20]. Although it is probably the oldest optimization method, it continues to play a central role in modern algorithmic theory and applications. Its definition can be written in a mere one line, x k 1 x k λ f (x k), k 0, (2) where x 0 R d is arbitrary and λ 0 .

We consider the combined use of resampling and partial rejection control in sequential Monte Carlo methods, also known as particle filters. While the variance reducing properties of rejection control are known, there has not been (to the best of our knowledge) any work on unbiased estimation of the marginal likelihood (also known as the model evidence or the normalizing constant) in this type of particle filters. Being able to estimate the marginal likelihood without bias is highly relevant for model comparison, computation of interpretable and reliable confidence intervals, and in exact approximation methods, such as particle Markov chain Monte Carlo. In the paper we present a particle filter with rejection control that enables unbiased estimation of the marginal likelihood.

Since manually labeling training data is slow and expensive, recent industrial and scientific research efforts have turned to weaker or noisier forms of supervision sources. However, existing weak supervision approaches fail to model multi-resolution sources for sequential data, like video, that can assign labels to individual elements or collections of elements in a sequence. A key challenge in weak supervision is estimating the unknown accuracies and correlations of these sources without using labeled data. Multi-resolution sources exacerbate this challenge due to complex correlations and sample complexity that scales in the length of the sequence. We propose Dugong, the first framework to model multi-resolution weak supervision sources with complex correlations to assign probabilistic labels to training data. Theoretically, we prove that Dugong, under mild conditions, can uniquely recover the unobserved accuracy and correlation parameters and use parameter sharing to improve sample complexity. Our method assigns clinician-validated labels to population-scale biomedical video repositories, helping outperform traditional supervision by 36.8 F1 points and addressing a key use case where machine learning has been severely limited by the lack of expert labeled data. On average, Dugong improves over traditional supervision by 16.0 F1 points and existing weak supervision approaches by 24.2 F1 points across several video and sensor classification tasks.

We propose a novel approach for sampling realistic financial correlation matrices. This approach is based on generative adversarial networks. Experiments demonstrate that generative adversarial networks are able to recover most of the known stylized facts about empirical correlation matrices estimated on asset returns. This is the first time such results are documented in the literature. Practical financial applications range from trading strategies enhancement to risk and portfolio stress testing. Such generative models can also help ground empirical finance deeper into science by allowing for falsifiability of statements and more objective comparison of empirical methods.

We consider the problem of tensor-response regression given covariates on multiple modes. Such data problems arise frequently in applications such as neuroimaging, network analysis, and spatial-temporal modeling. We propose a new family of tensor response regression models that incorporate covariates, and establish the theoretical accuracy guarantees. Unlike earlier methods, our estimation allows high-dimensionality in both the tensor response and the covariate matrices on multiple modes. An efficient alternating updating algorithm is further developed. Our proposal handles a broad range of data types, including continuous, count, and binary observations. Through simulation and applications to two real datasets, we demonstrate the outperformance of our approach over the state-of-art.

A graph homomorphism is a map between two graphs that preserves adjacency relations. We consider the problem of sampling a random graph homomorphism from a graph $F$ into a large network $\mathcal{G}$. When $\mathcal{G}$ is the complete graph with $q$ nodes, this becomes the well-known problem of sampling uniform $q$-colorings of $F$. We propose two complementary MCMC algorithms for sampling a random graph homomorphisms and establish bounds on their mixing times and concentration of their time averages. Based on our sampling algorithms, we propose a novel framework for network data analysis that circumvents some of the drawbacks in methods based on independent and neigborhood sampling. Various time averages of the MCMC trajectory give us real-, function-, and network-valued computable observables, including well-known ones such as homomorphism density and average clustering coefficient. One of the main observable we propose is called the conditional homomorphism density profile, which reveals hierarchical structure of the network. Furthermore, we show that these network observables are stable with respect to a suitably renormalized cut distance between networks. We also provide various examples and simulations demonstrating our framework through synthetic and real-world networks. For instance, we apply our framework to analyze Word Adjacency Networks of a 45 novels data set and propose an authorship attribution scheme using motif sampling and conditional homomorphism density profiles.

Large scale machine learning is increasingly relying on distributed optimization, whereby several machines contribute to the training process of a statistical model. While there exist a large literature on stochastic gradient descent (SGD) and variants, the study of countermeasures to mitigate problems arising in asynchronous distributed settings are still in their infancy. The key question of this work is whether sparsification, a technique predominantly used to reduce communication overheads, can also mitigate the staleness problem that affects asynchronous SGD. We study the role of sparsification both theoretically and empirically. Our theory indicates that, in an asynchronous, non-convex setting, the ergodic convergence rate of sparsified SGD matches the known result $\mathcal{O} \left( 1/\sqrt{T} \right)$ of non-convex SGD. We then carry out an empirical study to complement our theory and show that, in practice, sparsification consistently improves over vanilla SGD and current alternatives to mitigate the effects of staleness.

In the learning to learn (L2L) framework, we cast the design of optimization algorithms as a machine learning problem and use deep neural networks to learn the update rules. In this paper, we extend the L2L framework to zeroth-order (ZO) optimization setting, where no explicit gradient information is available. Our learned optimizer, modeled as recurrent neural network (RNN), first approximates gradient by ZO gradient estimator and then produces parameter update utilizing the knowledge of previous iterations. To reduce high variance effect due to ZO gradient estimator, we further introduce another RNN to learn the Gaussian sampling rule and dynamically guide the query direction sampling. Our learned optimizer outperforms hand-designed algorithms in terms of convergence rate and final solution on both synthetic and practical ZO optimization tasks (in particular, the black-box adversarial attack task, which is one of the most widely used tasks of ZO optimization). We finally conduct extensive analytical experiments to demonstrate the effectiveness of our proposed optimizer. Learning to learn (L2L) is a recently proposed meta-learning framework where we leverage deep neural networks to learn optimization algorithms automatically. The most common choice for the learned optimizer is recurrent neural network (RNN) since it can capture long-term dependencies and propose parameter updates based on knowledge of previous iterations. By training RNN op-timizers on predefined optimization tasks, the optimizers are capable of learning to explore the loss landscape and adaptively choose descent directions and steps (Lv et al., 2017).

Language creates a compact representation of the world and allows the description of unlimited situations and objectives through compositionality. These properties make it a natural fit to guide the training of interactive agents as it could ease recurrent challenges in Reinforcement Learning such as sample complexity, generalization, or multi-tasking. Yet, it remains an open-problem to relate language and RL in even simple instruction following scenarios. Current methods rely on expert demonstrations, auxiliary losses, or inductive biases in neural architectures. In this paper, we propose an orthogonal approach called Textual Hindsight Experience Replay (THER) that extends the Hindsight Experience Replay approach to the language setting. Whenever the agent does not fulfill its instruction, THER learns to output a new directive that matches the agent trajectory, and it relabels the episode with a positive reward. To do so, THER learns to map a state into an instruction by using past successful trajectories, which removes the need to have external expert interventions to relabel episodes as in vanilla HER. We observe that this simple idea also initiates a learning synergy between language acquisition and policy learning on instruction following tasks in the BabyAI environment.

While artificial intelligence (AI) and other automation technologies might lead to enormous progress in healthcare, they may also have undesired consequences for people working in the field. In this interdisciplinary study, we capture empirical evidence of not only what healthcare work could be automated, but also what should be automated. We quantitatively investigate these research questions by utilizing probabilistic machine learning models trained on thousands of ratings, provided by both healthcare practitioners and automation experts. Based on our findings, we present an analytical tool (Automatability-Desirability Matrix) to support policymakers and organizational leaders in developing practical strategies on how to harness the positive power of automation technologies, while accompanying change and empowering stakeholders in a participatory fashion.