We propose a computationally-friendly adaptive learning rate schedule, "AdaLoss", which directly uses the information of the loss function to adjust the stepsize in gradient descent methods. We prove that this schedule enjoys linear convergence in linear regression. Moreover, we provide a linear convergence guarantee over the non-convex regime, in the context of two-layer over-parameterized neural networks. If the width of the first-hidden layer in the two-layer networks is sufficiently large (polynomially), then AdaLoss converges robustly \emph{to the global minimum} in polynomial time. We numerically verify the theoretical results and extend the scope of the numerical experiments by considering applications in LSTM models for text clarification and policy gradients for control problems.
University of Oxford（牛津大学）；Auburn University（奥本大学）；Huawei Technologies（华为技术有限公司）
We present the first provably convergent two-timescale off-policy actor-critic algorithm (COF-PAC) with function approximation. Key to COF-PAC is the introduction of a new critic, the emphasis critic, which is trained via Gradient Emphasis Learning (GEM), a novel combination of the key ideas of Gradient Temporal Difference Learning and Emphatic Temporal Difference Learning. With the help of the emphasis critic and the canonical value function critic, we show convergence for COF-PAC, where the critics are linear and the actor can be nonlinear.
In Reinforcement Learning the Q-learning algorithm provably converges to the optimal solution. However, as others have demonstrated, Q-learning can also overestimate the values and thereby spend too long exploring unhelpful states. Double Q-learning is a provably convergent alternative that mitigates some of the overestimation issues, though sometimes at the expense of slower convergence. We introduce an alternative algorithm that replaces the max operation with an average, resulting also in a provably convergent off-policy algorithm which can mitigate overestimation yet retain similar converg…
Model-free learning-based control methods have seen great success recently. However, such methods typically suffer from poor sample complexity and limited convergence guarantees. This is in sharp contrast to classical model-based control, which has a rich theory but typically requires strong modeling assumptions. In this paper, we combine the two approaches to achieve the best of both worlds. We consider a dynamical system with both linear and non-linear components and develop a novel approach to use the linear model to define a warm start for a model-free, policy gradient method. We show this…
Iowa State University（爱荷华州立大学）；University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；New York University（纽约大学）；University of California Riverside（加州大学河滨分校）
The traditional approach of hand-crafting priors (such as sparsity) for solving inverse problems is slowly being replaced by the use of richer learned priors (such as those modeled by deep generative networks). In this work, we study the algorithmic aspects of such a learning-based approach from a theoretical perspective. For certain generative network architectures, we establish a simple non-convex algorithmic approach that (a) theoretically enjoys linear convergence guarantees for certain linear and nonlinear inverse problems, and (b) empirically improves upon conventional techniques such as…
MIT（麻省理工学院）；University of California, Berkeley（加州大学伯克利分校）
We introduce the first direct policy search algorithm which provably converges to the globally optimal $\textit{dynamic}$ filter for the classical problem of predicting the outputs of a linear dynamical system, given noisy, partial observations. Despite the ubiquity of partial observability in practice, theoretical guarantees for direct policy search algorithms, one of the backbones of modern reinforcement learning, have proven difficult to achieve. This is primarily due to the degeneracies which arise when optimizing over filters that maintain internal state. In this paper, we provide a new p…
Univ. de Rouen（鲁昂大学）；Criteo AI Lab（Criteo人工智能实验室）；Univ Côte d’Azur（蔚蓝海岸大学）；CNRS（法国国家科学研究中心）；OCA Lagrange（拉格朗日蔚蓝海岸天文台）；INSA de Rouen（鲁昂国立应用科学学院）；IMAG（蒙彼利埃亚历山大·格罗滕迪克研究所）；Université de Montpellier（蒙彼利埃大学）
Owing to their statistical properties, non-convex sparse regularizers have attracted much interest for estimating a sparse linear model from high dimensional data. Given that the solution is sparse, for accelerating convergence, a working set strategy addresses the optimization problem through an iterative algorithm by incre-menting the number of variables to optimize until the identification of the solution support. While those methods have been well-studied and theoretically supported for convex regularizers, this paper proposes a working set algorithm for non-convex sparse regularizers with…
Rice University（莱斯大学）；IBM T.J. Watson Research Center（IBM T.J. 沃森研究中心）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
We present theoretical results on the convergence of \emph{non-convex} accelerated gradient descent in matrix factorization models with $\ell_2$-norm loss. The purpose of this work is to study the effects of acceleration in non-convex settings, where provable convergence with acceleration should not be considered a \emph{de facto} property. The technique is applied to matrix sensing problems, for the estimation of a rank $r$ optimal solution $X^\star \in \mathbb{R}^{n \times n}$. Our contributions can be summarized as follows. $i)$ We show that acceleration in factored gradient descent converg…
Exact inference in the linear regression model with spike and slab priors is often intractable. Expectation propagation (EP) can be used for approximate inference. However, the regular sequential form of EP (R-EP) may fail to converge in this model when the size of the training set is very small. As an alternative, we propose a provably convergent EP algorithm (PC-EP). PC-EP is proved to minimize an energy function which, under some constraints, is bounded from below and whose stationary points coincide with the solution of R-EP. Experiments with synthetic data indicate that when R-EP does not…
University of Oxford（牛津大学）；London School of Economics and Political Science（伦敦政治经济学院）
Despite its popularity in the reinforcement learning community, a provably convergent policy gradient method for continuous space-time control problems with nonlinear state dynamics has been elusive. This paper proposes proximal gradient algorithms for feedback controls of finite-time horizon stochastic control problems. The state dynamics are nonlinear diffusions with control-affine drift, and the cost functions are nonconvex in the state and nonsmooth in the control. The system noise can degenerate, which allows for deterministic control problems as special cases. We prove under suitable con…
University of California, Berkeley（加利福尼亚大学伯克利分校）；Amazon（亚马逊）
Linear dynamical systems are a fundamental and powerful parametric model class. However, identifying the parameters of a linear dynamical system is a venerable task, permitting provably efficient solutions only in special cases. This work shows that the eigenspectrum of unknown linear dynamics can be identified without full system identification. We analyze a computationally efficient and provably convergent algorithm to estimate the eigenvalues of the state-transition matrix in a linear dynamical system. When applied to time series clustering, our algorithm can efficiently cluster multi-dimen…
We propose a family of adaptive integer compression operators for distributed Stochastic Gradient Descent (SGD) that do not communicate a single float. This is achieved by multiplying floating-point vectors with a number known to every device and then rounding to integers. In contrast to the prior work on integer compression for SwitchML by Sapio et al. (2021), our IntSGD method is provably convergent and computationally cheaper as it estimates the scaling of vectors adaptively. Our theory shows that the iteration complexity of IntSGD matches that of SGD up to constant factors for both convex …
We study infinite-horizon discounted two-player zero-sum Markov games, and develop a decentralized algorithm that provably converges to the set of Nash equilibria under self-play. Our algorithm is based on running an Optimistic Gradient Descent Ascent algorithm on each state to learn the policies, with a critic that slowly learns the value of each state. To the best of our knowledge, this is the first algorithm in this setting that is simultaneously rational (converging to the opponent's best response when it uses a stationary policy), convergent (converging to the set of Nash equilibria under…
We analyze two novel randomized variants of the Frank-Wolfe (FW) or conditional gradient algorithm. While classical FW algorithms require solving a linear minimization problem over the domain at each iteration, the proposed method only requires to solve a linear minimization problem over a small \emph{subset} of the original domain. The first algorithm that we propose is a randomized variant of the original FW algorithm and achieves a $\mathcal{O}(1/t)$ sublinear convergence rate as in the deterministic counterpart. The second algorithm is a randomized variant of the Away-step FW algorithm, an…
We consider the class of optimization problems arising from computationally intensive L1-regularized M-estimators, where the function or gradient values are very expensive to compute. A particular instance of interest is the L1-regularized MLE for learning Conditional Random Fields (CRFs), which are a popular class of statistical models for varied structured prediction problems such as sequence labeling, alignment, and classification with label taxonomy. L1-regularized MLEs for CRFs are particularly expensive to optimize since computing the gradient values requires an expensive inference step.…
