We revisit the use of Stochastic Gradient Descent (SGD) for solving convex optimization problems that serve as highly popular convex relaxations for many important low-rank matrix recovery problems such as \textit{matrix completion}, \textit{phase retrieval}, and more. The computational limitation of applying SGD to solving these relaxations in large-scale is the need to compute a potentially high-rank singular value decomposition (SVD) on each iteration in order to enforce the low-rank-promoting constraint. We begin by considering a simple and natural sufficient condition so that these relaxations indeed admit low-rank solutions. This condition is also necessary for a certain notion of low-rank-robustness to hold. Our main result shows that under this condition which involves the eigenvalues of the gradient vector at optimal points, SGD with mini-batches, when initialized with a "warm-start" point, produces iterates that are low-rank with high probability, and hence only a low-rank SVD computation is required on each iteration. This suggests that SGD may indeed be practically applicable to solving large-scale convex relaxations of low-rank matrix recovery problems. Our theoretical results are accompanied with supporting preliminary empirical evidence. As a side benefit, our analysis is quite simple and short.

In Machine Learning and Optimization community there are two main approaches for convex risk minimization problem. The first approach is Stochastic Averaging (SA) (online) and the second one is Stochastic Average Approximation (SAA) (Monte Carlo, Empirical Risk Minimization, offline) with proper regularization in non-strongly convex case. At the moment, it is known that both approaches are on average equivalent (up to a logarithmic factor) in terms of oracle complexity (required number of stochastic gradient evaluations). What is the situation with total complexity? The answer depends on specific problem. However, starting from work [Nemirovski et al. (2009)] it was generally accepted that SA is better than SAA. Nevertheless, in case of large-scale problems SA may ran out of memory problems since storing all data on one machine and organizing online access to it can be impossible without communications with other machines. SAA in contradistinction to SA allows parallel/distributed calculations. In this paper we show that SAA may outperform SA in the problem of calculating an estimation for population ({\mu}-entropy regularized) Wasserstein barycenter even for non-parallel (non-decenralized) set up.

In many online learning problems the computational bottleneck for gradient-based methods is the projection operation. For this reason, in many problems the most efficient algorithms are based on the Frank-Wolfe method, which replaces projections by linear optimization. In the general case, however, online projection-free methods require more iterations than projection-based methods: the best known regret bound scales as $T^{3/4}$. Despite significant work on various variants of the Frank-Wolfe method, this bound has remained unchanged for a decade. In this paper we give an efficient projection-free algorithm that guarantees $T^{2/3}$ regret for general online convex optimization with smooth cost functions and one linear optimization computation per iteration. As opposed to previous Frank-Wolfe approaches, our algorithm is derived using the Follow-the-Perturbed-Leader method and is analyzed using an online primal-dual framework.

--Deep neural networks achieve state-of-the-art performance for a range of classification and inference tasks. However, the use of stochastic gradient descent combined with the noncon-vexity of the underlying optimization problems renders parameter learning susceptible to initialization. T o address this issue, a variety of methods that rely on random parameter initialization or knowledge distillation have been proposed in the past. In this paper, we propose FuseInit, a novel method to initialize shallower networks by fusing neighboring layers of deeper networks that are trained with random initialization. We develop theoretical results and efficient algorithms for mean-square error (MSE)- optimal fusion of neighboring dense-dense, convolutional-dense, and convolutional-convolutional layers. We show experiments for a range of classification and regression datasets, which suggest that deeper neural networks are less sensitive to initialization and shallower networks can perform better (sometimes as well as their deeper counterparts) if initialized with FuseInit.

This short note reviews so-called Natural Gradient Descent (NGD) for multivariate Gaussians. The Fisher Information Matrix (FIM) is derived for several different parameterizations of Gaussians. Careful attention is paid to the symmetric nature of the covariance matrix when calculating derivatives. We show that there are some advantages to choosing a parameterization comprising the mean and inverse covariance matrix and provide a simple NGD update that accounts for the symmetric (and sparse) nature of the inverse covariance matrix.

The article considers smooth optimization of functions on Lie groups. By generalizing NAG variational principle in vector space (Wibisono et al., 2016) to Lie groups, continuous Lie-NAG dynamics which are guaranteed to converge to local optimum are obtained. They correspond to momentum versions of gradient flow on Lie groups. A particular case of $\mathsf{SO}(n)$ is then studied in details, with objective functions corresponding to leading Generalized EigenValue problems: the Lie-NAG dynamics are first made explicit in coordinates, and then discretized in structure preserving fashions, resulting in optimization algorithms with faithful energy behavior (due to conformal symplecticity) and exactly remaining on the Lie group. Stochastic gradient versions are also investigated. Numerical experiments on both synthetic data and practical problem (LDA for MNIST) demonstrate the effectiveness of the proposed methods as optimization algorithms ($not$ as a classification method).

Authors suggest their method is a natural generalization of gradient descent to the two-player scenario where the update is given by the Nash equilibrium of a regularized bilinear local approximation of the underlying game. It avoids oscillatory and divergent behaviors seen in alternating gradient descent. The paper proposes several experiments to establish the robustness of their method. This project aims at replicating their results. The paper provides a detailed comparison to methods based on optimism and consensus on the properties of convergence and stability of various discussed methods using numerical experiments and rigorous analysis. In order to understand these terms, comparison and proposed method and examine the results of the experiments, next section gives a necessary background of the original paper. 2 Background The traditional optimization is concerned with a single agent trying to optimize a cost function. It can be seen as min x R m f ( x). The agent has a clear objective to find ("Good local") minimum of f . Gradeint Descent (and its varients) are reliable Algorithmic Baseline for this purpose.

We consider the problem of controlling a possibly unknown linear dynamical system with adversarial perturbations, adversarially chosen convex loss functions, and partially observed states, known as non-stochastic control. We introduce a controller parametrization based on the denoised observations, and prove that applying online gradient descent to this parametrization yields a new controller which attains sublinear regret vs. a large class of closed-loop policies. In the fully-adversarial setting, our controller attains an optimal regret bound of $\sqrt{T}$-when the system is known, and, when combined with an initial stage of least-squares estimation, $T^{2/3}$ when the system is unknown; both yield the first sublinear regret for the partially observed setting. Our bounds are the first in the non-stochastic control setting that compete with \emph{all} stabilizing linear dynamical controllers, not just state feedback. Moreover, in the presence of semi-adversarial noise containing both stochastic and adversarial components, our controller attains the optimal regret bounds of $\mathrm{poly}(\log T)$ when the system is known, and $\sqrt{T}$ when unknown. To our knowledge, this gives the first end-to-end $\sqrt{T}$ regret for online Linear Quadratic Gaussian controller, and applies in a more general setting with adversarial losses and semi-adversarial noise.

Our main result concerns the following condition: {\bf Condition C.} Let $X$ be a Banach space. A $C^1$ function $f:X\rightarrow \mathbb{R}$ satisfies Condition C if whenever $\{x_n\}$ weakly converges to $x$ and $\lim _{n\rightarrow\infty}||\nabla f(x_n)||=0$, then $\nabla f(x)=0$. We assume that there is given a canonical isomorphism between $X$ and its dual $X^*$, for example when $X$ is a Hilbert space. {\bf Theorem.} Let $X$ be a reflexive, complete Banach space and $f:X\rightarrow \mathbb{R}$ be a $C^2$ function which satisfies Condition C. Moreover, we assume that for every bounded set $S\subset X$, then $\sup _{x\in S}||\nabla ^2f(x)||<\infty$. We choose a random point $x_0\in X$ and construct by the Local Backtracking GD procedure (which depends on $3$ hyper-parameters $\alpha ,\beta ,\delta _0$, see later for details) the sequence $x_{n+1}=x_n-\delta (x_n)\nabla f(x_n)$. Then we have: 1) Every cluster point of $\{x_n\}$, in the {\bf weak} topology, is a critical point of $f$. 2) Either $\lim _{n\rightarrow\infty}f(x_n)=-\infty$ or $\lim _{n\rightarrow\infty}||x_{n+1}-x_n||=0$. 3) Here we work with the weak topology. Let $\mathcal{C}$ be the set of critical points of $f$. Assume that $\mathcal{C}$ has a bounded component $A$. Let $\mathcal{B}$ be the set of cluster points of $\{x_n\}$. If $\mathcal{B}\cap A\not= \emptyset$, then $\mathcal{B}\subset A$ and $\mathcal{B}$ is connected. 4) Assume that $X$ is separable. Then for generic choices of $\alpha ,\beta ,\delta _0$ and the initial point $x_0$, if the sequence $\{x_n\}$ converges - in the {\bf weak} topology, then the limit point cannot be a saddle point.

Jing Dong and Xin T. Tong † January 24, 2020 Abstract Gradient descent (GD) is known to converge quickly for convex objective functions, but it can be trapped at local minimums. On the other hand, Langevin dynamics (LD) can explore the state space and find global minimums, but in order to give accurate estimates, LD needs to run with small discretization stepsize and weak stochastic force, which in general slow down its convergence. This paper shows that these two algorithms can "collaborate" through a simple exchange mechanism, in which they swap their current positions if LD yields a lower objective function. This idea can be seen as the singular limit of the replica exchange technique from the sampling literature. We show that this new algorithm converges to the global minimum linearly with high probability, assuming the objective function is strongly convex in a neighborhood of the unique global minimum. By replacing gradients with stochastic gradients, and adding a proper threshold to the exchange mechanism, our algorithm can also be used in online settings. We further verify our theoretical results through some numerical experiments, and observe superior performance of the proposed algorithm over running GD or LD alone. 1 Introduction Division of labor is the secret of any efficient enterprises. By collaborating with individuals with different skillsets, we can focus on tasks within our own expertise and produce better outcomes than working independently. This paper asks whether the same principle can be applied when designing an algorithm. Given a general smooth non-convex objective function F, we consider the unconstrained optimization problem min x R dF ( x). However, this local minimum may not be the global minimum, and GD will be trapped there afterwards. On the other hand, sampling-based algorithms, such as the Langevin dynamics (LD) can escape local minimums by their stochasticity, but the additional stochastic noise contaminates the optimization results and slows down the convergence when the iterate is near the global minimum. In general, deterministic algorithms are designed to finding local minimums quickly, but they can be terrible in exploration. Sampling-based algorithms are better suited for exploring the state space, but they are inefficient when pinpointing the local minimums. This paper investigates how they can "collaborate" to get the "best of the two worlds". The collaboration mechanism we introduced here comes from replica-exchange in the sampling literature. Its implementation is very simple: we run a copy of GD, denoted by X n; and a copy of LD, denoted by Y n. If F (X n) F (Y n), we swap their positions.