We study stochastic gradient descent for solving conditional stochastic optimization problems, in which an objective to be minimized is given by a parametric nested expectation with an outer expectation taken with respect to one random variable and an inner conditional expectation with respect to the other random variable. The gradient of such a parametric nested expectation is again expressed as a nested expectation, which makes it hard for the standard nested Monte Carlo estimator to be unbiased. In this paper, we show under some conditions that a multilevel Monte Carlo gradient estimator is unbiased and has finite variance and finite expected computational cost, so that the standard theory from stochastic optimization for a parametric (non-nested) expectation directly applies. We also discuss a special case for which yet another unbiased gradient estimator with finite variance and cost can be constructed.

Off-policy algorithms, in which a behavior policy differs from the target policy and is used to gain experience for learning, have proven to be of great practical value in reinforcement learning. However, even for simple convex problems such as linear value function approximation, these algorithms are not guaranteed to be stable. To address this, alternative algorithms that are provably convergent in such cases have been introduced, the most well known being gradient descent temporal difference (GTD) learning. This algorithm and others like it, however, tend to converge much more slowly than conventional temporal difference learning. In this paper we propose gradient descent temporal difference-difference (Gradient-DD) learning in order to improve GTD2, a GTD algorithm, by introducing second-order differences in successive parameter updates. We investigate this algorithm in the framework of linear value function approximation, theoretically proving its convergence by applying the theory of stochastic approximation. %analytically showing its improvement over GTD2. Studying the model empirically on the random walk task, the Boyan-chain task, and the Baird's off-policy counterexample, we find substantial improvement over GTD2 and, in several cases, better performance even than conventional TD learning.

This work studies constrained stochastic optimization problems where the objective and constraint functions are convex and expressed as compositions of stochastic functions. The problem arises in the context of fair classification, fair regression, and the design of queuing systems. Of particular interest is the large-scale setting where an oracle provides the stochastic gradients of the constituent functions, and the goal is to solve the problem with a minimal number of calls to the oracle. Owing to the compositional form, the stochastic gradients provided by the oracle do not yield unbiased estimates of the objective or constraint gradients. Instead, we construct approximate gradients by tracking the inner function evaluations, resulting in a quasi-gradient saddle point algorithm. We prove that the proposed algorithm is guaranteed to find the optimal and feasible solution almost surely. We further establish that the proposed algorithm requires $\mathcal{O}(1/\epsilon^4)$ data samples in order to obtain an $\epsilon$-approximate optimal point while also ensuring zero constraint violation. The result matches the sample complexity of the stochastic compositional gradient descent method for unconstrained problems and improves upon the best-known sample complexity results for the constrained settings. The efficacy of the proposed algorithm is tested on both fair classification and fair regression problems. The numerical results show that the proposed algorithm outperforms the state-of-the-art algorithms in terms of the convergence rate.

The aim of Few-Shot learning methods is to train models which can easily adapt to previously unseen tasks, based on small amounts of data. One of the most popular and elegant Few-Shot learning approaches is Model-Agnostic Meta-Learning (MAML). The main idea behind this method is to learn the general weights of the meta-model, which are further adapted to specific problems in a small number of gradient steps. However, the model's main limitation lies in the fact that the update procedure is realized by gradient-based optimisation. In consequence, MAML cannot always modify weights to the essential level in one or even a few gradient iterations. On the other hand, using many gradient steps results in a complex and time-consuming optimization procedure, which is hard to train in practice, and may lead to overfitting. In this paper, we propose HyperMAML, a novel generalization of MAML, where the training of the update procedure is also part of the model. Namely, in HyperMAML, instead of updating the weights with gradient descent, we use for this purpose a trainable Hypernetwork. Consequently, in this framework, the model can generate significant updates whose range is not limited to a fixed number of gradient steps. Experiments show that HyperMAML consistently outperforms MAML and performs comparably to other state-of-the-art techniques in a number of standard Few-Shot learning benchmarks.

The training of deep neural networks and other modern machine learning models usually consists in solving non-convex optimisation problems that are high-dimensional and subject to large-scale data. Here, momentum-based stochastic optimisation algorithms have become especially popular in recent years. The stochasticity arises from data subsampling which reduces computational cost. Moreover, both, momentum and stochasticity are supposed to help the algorithm to overcome local minimisers and, hopefully, converge globally. Theoretically, this combination of stochasticity and momentum is badly understood. In this work, we propose and analyse a continuous-time model for stochastic gradient descent with momentum. This model is a piecewise-deterministic Markov process that represents the particle movement by an underdamped dynamical system and the data subsampling through a stochastic switching of the dynamical system. In our analysis, we investigate longtime limits, the subsampling-to-no-subsampling limit, and the momentum-to-no-momentum limit. We are particularly interested in the case of reducing the momentum over time: intuitively, the momentum helps to overcome local minimisers in the initial phase of the algorithm, but prohibits fast convergence to a global minimiser later. Under convexity assumptions, we show convergence of our dynamical system to the global minimiser when reducing momentum over time and let the subsampling rate go to infinity. We then propose a stable, symplectic discretisation scheme to construct an algorithm from our continuous-time dynamical system. In numerical experiments, we study our discretisation scheme in convex and non-convex test problems. Additionally, we train a convolutional neural network to solve the CIFAR-10 image classification problem. Here, our algorithm reaches competitive results compared to stochastic gradient descent with momentum.

Federated learning (FL) has attracted much attention as a privacy-preserving distributed machine learning framework, where many clients collaboratively train a machine learning model by exchanging model updates with a parameter server instead of sharing their raw data. Nevertheless, FL training suffers from slow convergence and unstable performance due to stragglers caused by the heterogeneous computational resources of clients and fluctuating communication rates. This paper proposes a coded FL framework to mitigate the straggler issue, namely stochastic coded federated learning (SCFL). In this framework, each client generates a privacy-preserving coded dataset by adding additive noise to the random linear combination of its local data. The server collects the coded datasets from all the clients to construct a composite dataset, which helps to compensate for the straggling effect. In the training process, the server as well as clients perform mini-batch stochastic gradient descent (SGD), and the server adds a make-up term in model aggregation to obtain unbiased gradient estimates. We characterize the privacy guarantee by the mutual information differential privacy (MI-DP) and analyze the convergence performance in federated learning. Besides, we demonstrate a privacy-performance tradeoff of the proposed SCFL method by analyzing the influence of the privacy constraint on the convergence rate. Finally, numerical experiments corroborate our analysis and show the benefits of SCFL in achieving fast convergence while preserving data privacy.

In this paper, we propose a novel accelerated gradient method called ANITA for solving the fundamental finite-sum optimization problems. Concretely, we consider both general convex and strongly convex settings: i) For general convex finite-sum problems, ANITA improves previous state-of-the-art result given by Varag (Lan et al., 2019). In particular, for large-scale problems or the convergence error is not very small, i.e., $n \geq \frac{1}{\epsilon^2}$, ANITA obtains the \emph{first} optimal result $O(n)$, matching the lower bound $\Omega(n)$ provided by Woodworth and Srebro (2016), while previous results are $O(n \log \frac{1}{\epsilon})$ of Varag (Lan et al., 2019) and $O(\frac{n}{\sqrt{\epsilon}})$ of Katyusha (Allen-Zhu, 2017). ii) For strongly convex finite-sum problems, we also show that ANITA can achieve the optimal convergence rate $O\big((n+\sqrt{\frac{nL}{\mu}})\log\frac{1}{\epsilon}\big)$ matching the lower bound $\Omega\big((n+\sqrt{\frac{nL}{\mu}})\log\frac{1}{\epsilon}\big)$ provided by Lan and Zhou (2015). Besides, ANITA enjoys a simpler loopless algorithmic structure unlike previous accelerated algorithms such as Varag (Lan et al., 2019) and Katyusha (Allen-Zhu, 2017) where they use double-loop structures. Moreover, we provide a novel \emph{dynamic multi-stage convergence analysis}, which is the key technical part for improving previous results to the optimal rates. We believe that our new theoretical rates and novel convergence analysis for the fundamental finite-sum problem will directly lead to key improvements for many other related problems, such as distributed/federated/decentralized optimization problems (e.g., Li and Richt\'arik, 2021). Finally, the numerical experiments show that ANITA converges faster than the previous state-of-the-art Varag (Lan et al., 2019), validating our theoretical results and confirming the practical superiority of ANITA.

Riemannian optimization is a principled framework for solving optimization problems where the desired optimum is constrained to a smooth manifold $\mathcal{M}$. Algorithms designed in this framework usually require some geometrical description of the manifold, which typically includes tangent spaces, retractions, and gradients of the cost function. However, in many cases, only a subset (or none at all) of these elements can be accessed due to lack of information or intractability. In this paper, we propose a novel approach that can perform approximate Riemannian optimization in such cases, where the constraining manifold is a submanifold of $\R^{D}$. At the bare minimum, our method requires only a noiseless sample set of the cost function $(\x_{i}, y_{i})\in {\mathcal{M}} \times \mathbb{R}$ and the intrinsic dimension of the manifold $\mathcal{M}$. Using the samples, and utilizing the Manifold-MLS framework (Sober and Levin 2020), we construct approximations of the missing components entertaining provable guarantees and analyze their computational costs. In case some of the components are given analytically (e.g., if the cost function and its gradient are given explicitly, or if the tangent spaces can be computed), the algorithm can be easily adapted to use the accurate expressions instead of the approximations. We analyze the global convergence of Riemannian gradient-based methods using our approach, and we demonstrate empirically the strength of this method, together with a conjugate-gradients type method based upon similar principles.

The robustness of any machine learning solution is fundamentally bound by the data it was trained on. One way to generalize beyond the original training is through human-informed augmentation of the original dataset; however, it is impossible to specify all possible failure cases that can occur during deployment. To address this limitation we combine model-based reinforcement learning and model-interpretability methods to propose a solution that self-generates simulated scenarios constrained by environmental concepts and dynamics learned in an unsupervised manner. In particular, an internal model of the agent's environment is conditioned on low-dimensional concept representations of the input space that are sensitive to the agent's actions. We demonstrate this method within a standard realistic driving simulator in a simple point-to-point navigation task, where we show dramatic improvements in one-shot generalization to different instances of specified failure cases as well as zero-shot generalization to similar variations compared to model-based and model-free approaches.

Abstract: A common way to train neural networks is the Backpropagation. This algorithm includes a gradient descent method, which needs an adaptive step size. In the area of neural networks, the ADAM-Optimizer is one of the most popular adaptive step size methods. The 5865 citations in only three years shows additionally the importance of the given paper. We discovered that the given convergence proof of the optimizer contains some mistakes, so that the proof will be wrong.