The multinomial logistic regression (MLR) model is widely used in statistics and machine learning. Stochastic gradient descent (SGD) is the most common approach for determining the parameters of a MLR model in big data scenarios. However, SGD has slow sub-linear rates of convergence. A way to improve these rates of convergence is to use manifold optimization. Along this line, stochastic natural gradient descent (SNGD), proposed by Amari, was proven to be Fisher efficient when it converged. However, SNGD is not guaranteed to converge and it is computationally too expensive for MLR models with a large number of parameters.
Here, we propose a stochastic optimization method for MLR based on manifold optimization concepts which (i) has per-iteration computational complexity is linear in the number of parameters and (ii) can be proven to converge.
To achieve (i) we establish that the family of joint distributions for MLR is a dually flat manifold and we use that to speed up calculations. Sánchez-López and Cerquides have recently introduced convergent stochastic natural gradient descent (CSNGD), a variant of SNGD whose convergence is guaranteed. To obtain (ii) our algorithm uses the fundamental idea from CSNGD, thus relying on an independent sequence to build a bounded approximation of the natural gradient. We call the resulting algorithm dual stochastic natural gradient descent (DNSGD). By generalizing a result from Sunehag et al., we prove that DSNGD converges. Furthermore, we prove that the computational complexity of DSNGD iterations are linear on the number of variables of the model.
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We analyze the properties of gradient descent on convex surrogates for the zero-one loss for the agnostic learning of linear halfspaces. If $\mathsf{OPT}$ is the best classification error achieved by a halfspace, by appealing to the notion of soft margins we are able to show that gradient descent finds halfspaces with classification error $\tilde O(\mathsf{OPT}^{1/2}) + \varepsilon$ in $\mathrm{poly}(d,1/\varepsilon)$ time and sample complexity for a broad class of distributions that includes log-concave isotropic distributions as a subclass. Along the way we answer a question recently posed b…
The University of Tokyo（东京大学）；RIKEN（理化学研究所）；Japan Science and Technology Agency（日本科学技术振兴机构）
The superior performance of ensemble methods with infinite models are well known. Most of these methods are based on optimization problems in infinite-dimensional spaces with some regularization, for instance, boosting methods and convex neural networks use $L^1$-regularization with the non-negative constraint. However, due to the difficulty of handling $L^1$-regularization, these problems require early stopping or a rough approximation to solve it inexactly. In this paper, we propose a new ensemble learning method that performs in a space of probability measures, that is, our method can handl…
Nagoya Institute of Technology（名古屋工业大学）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
Deep unfolding is a promising deep-learning technique, whose network architecture is based on expanding the recursive structure of existing iterative algorithms. Although convergence acceleration is a remarkable advantage of deep unfolding, its theoretical aspects have not been revealed yet. The first half of this study details the theoretical analysis of the convergence acceleration in deep-unfolded gradient descent (DUGD) whose trainable parameters are step sizes. We propose a plausible interpretation of the learned step-size parameters in DUGD by introducing the principle of Chebyshev steps…
Variational methods are widely applied to ill-posed inverse problems for they have the ability to embed prior knowledge about the solution. However, the level of performance of these methods significantly depends on a set of parameters, which can be estimated through computationally expensive and time-consuming methods. In contrast, deep learning offers very generic and efficient architectures, at the expense of explainability, since it is often used as a black-box, without any fine control over its output. Deep unfolding provides a convenient approach to combine variational-based and deep lea…
In this work, we consider the resilience of distributed algorithms based on stochastic gradient descent (SGD) in distributed learning with potentially Byzantine attackers, who could send arbitrary information to the parameter server to disrupt the training process. Toward this end, we propose a new Lipschitz-inspired coordinate-wise median approach (LICM-SGD) to mitigate Byzantine attacks. We show that our LICM-SGD algorithm can resist up to half of the workers being Byzantine attackers, while still converging almost surely to a stationary region in non-convex settings. Also, our LICM-SGD meth…
Given recent deep learning results that demonstrate the ability to effectively optimize high-dimensional non-convex functions with gradient descent optimization on GPUs, we ask in this paper whether symbolic gradient optimization tools such as Tensorflow can be effective for planning in hybrid (mixed discrete and continuous) nonlinear domains with high dimensional state and action spaces? To this end, we demonstrate that hybrid planning with Tensorflow and RMSProp gradient descent is competitive with mixed integer linear program (MILP) based optimization on piecewise linear planning domains (w…
University of California, Berkeley（加利福尼亚大学伯克利分校）；Toyota Technological Institute at Chicago（芝加哥丰田理工学院）
Adaptive optimization methods, which perform local optimization with a metric constructed from the history of iterates, are becoming increasingly popular for training deep neural networks. Examples include AdaGrad, RMSProp, and Adam. We show that for simple overparameterized problems, adaptive methods often find drastically different solutions than gradient descent (GD) or stochastic gradient descent (SGD). We construct an illustrative binary classification problem where the data is linearly separable, GD and SGD achieve zero test error, and AdaGrad, Adam, and RMSProp attain test errors arbitr…
The loss surface of deep neural networks has recently attracted interest in the optimization and machine learning communities as a prime example of high-dimensional non-convex problem. Some insights were recently gained using spin glass models and mean-field approximations, but at the expense of strongly simplifying the nonlinear nature of the model. In this work, we do not make any such assumption and study conditions on the data distribution and model architecture that prevent the existence of bad local minima. Our theoretical work quantifies and formalizes two important \emph{folklore} fact…
Checkable NTK Positivity and Finite-Width Gradient Descent for Scalar- and Vector-Valued PINNs with Strong-Form, Weak-Form, and Nonlocal Linear Constraints
Blackwell approachability is a framework for reasoning about repeated games with vector-valued payoffs. We introduce predictive Blackwell approachability, where an estimate of the next payoff vector is given, and the decision maker tries to achieve better performance based on the accuracy of that estimator. In order to derive algorithms that achieve predictive Blackwell approachability, we start by showing a powerful connection between four well-known algorithms. Follow-the-regularized-leader (FTRL) and online mirror descent (OMD) are the most prevalent regret minimizers in online convex optim…
Generalizing the simplex method, circuit augmentation schemes for linear programs follow circuit directions through the interior of the underlying polyhedron. Steepest-descent augmentation is especially promising, but an implementation of the iterative scheme is a significant challenge. We work towards a viable implementation through a model in which a single linear program is updated dynamically to remain in memory throughout. Computational experiments exhibit dramatic improvements over a naive approach and reveal insight into the next steps required for large-scale computations.
School of Science, Sun Yat-sen University（中山大学理学院）；Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong（香港中文大学系统工程及工程管理学系）；School of Mathematical Sciences, Xiamen University（厦门大学数学科学学院）
We propose a first-order method for solving inequality constrained optimization problems. The method is derived from our previous work [12], a modified search direction method (MSDM) that applies the singular-value decomposition of normalized gradients. In this work, we simplify its computational framework to a "gradient descent akin" method, i.e., the search direction is computed using a linear combination of the negative and normalized objective and constraint gradient. The main focus of this work is to provide a mathematical aspect to the method. We analyze the global behavior and convergen…
In this paper, we develop an $O((m \log k) {\rm MSF} (n,m,1))$-time algorithm to find a half-integral node-capacitated multiflow of the maximum total flow-value in a network with $n$ nodes, $m$ edges, and $k$ terminals, where ${\rm MSF} (n',m',γ)$ denotes the time complexity of solving the maximum submodular flow problem in a network with $n'$ nodes, $m'$ edges, and the complexity $γ$ of computing the exchange capacity of the submodular function describing the problem. By using Fujishige-Zhang algorithm for submodular flow, we can find a maximum half-integral multiflow in $O(m n^3 \log k)$ tim…
We consider online convex optimization with stochastic constraints where the objective functions are arbitrarily time-varying and the constraint functions are independent and identically distributed (i.i.d.) over time. Both the objective and constraint functions are revealed after the decision is made at each time slot. The best known expected regret for solving such a problem is $\mathcal{O}(\sqrt{T})$, with a coefficient that is polynomial in the dimension of the decision variable and relies on the Slater condition (i.e. the existence of interior point assumption), which is restrictive and i…
We develop a new interior-point method (IPM) for symmetric-cone optimization, a common generalization of linear, second-order-cone, and semidefinite programming. In contrast to classical IPMs, we update iterates with a geodesic of the cone instead of the kernel of the linear constraints. This approach yields a primal-dual-symmetric, scale-invariant, and line-search-free algorithm that uses just half the variables of a standard primal-dual IPM. With elementary arguments, we establish polynomial-time convergence matching the standard square-root-n bound. Finally, we prove global convergence of a…
This paper presents a numerical algorithm for computing 6-degree-of-freedom free-final-time powered descent guidance trajectories. The trajectory generation problem is formulated using a unit dual quaternion representation of the rigid body dynamics, and several standard path constraints. Our formulation also includes a special line of sight constraints that is enforced only within a specified band of slant ranges relative to the landing site, a novel feature that is especially relevant to Terrain and Hazard Relative Navigation. We use the newly introduced state-triggered constraints to formul…
We provide new tools for worst-case performance analysis of the gradient (or steepest descent) method of Cauchy for smooth strongly convex functions, and Newton's method for self-concordant functions, including the case of inexact search directions. The analysis uses semidefinite programming performance estimation, as pioneered by Drori and Teboulle [Mathematical Programming, 145(1-2):451-482, 2014], and extends recent performance estimation results for the method of Cauchy by the authors [Optimization Letters, 11(7), 1185-1199, 2017]. To illustrate the applicability of the tools, we demonstra…
In this paper we provide faster algorithms for solving the geometric median problem: given $n$ points in $\mathbb{R}^{d}$ compute a point that minimizes the sum of Euclidean distances to the points. This is one of the oldest non-trivial problems in computational geometry yet despite an abundance of research the previous fastest algorithms for computing a $(1+ε)$-approximate geometric median were $O(d\cdot n^{4/3}ε^{-8/3})$ by Chin et. al, $\tilde{O}(d\exp{ε^{-4}\logε^{-1}})$ by Badoiu et. al, $O(nd+\mathrm{poly}(d,ε^{-1})$ by Feldman and Langberg, and $O((nd)^{O(1)}\log\frac{1}ε)$ by Parrilo a…
Interior point methods solve small to medium sized problems to high accuracy in a reasonable amount of time. However, for larger problems as well as stochastic problems, one needs to use first-order methods such as stochastic gradient descent (SGD), the alternating direction method of multipliers (ADMM), and conjugate gradient (CG) in order to attain a modest accuracy in a reasonable number of iterations. In this report, we first discuss heuristics for diagonal scaling. Next, we motivate preconditioning by an example, and then we study preconditioning for a specific splitting form in ADMM call…
