arXiv subjects

Martin Takáč

Publications and source records attributed to Martin Takáč.

At least 127 records · Page 7Linked to original sources

arXiv2020

Alternating Maximization: Unifying Framework for 8 Sparse PCA Formulations and Efficient Parallel Codes

Given a multivariate data set, sparse principal component analysis (SPCA) aims to extract several linear combinations of the variables that together explain the variance in the data as much as possible, while controlling the number of nonzero loadings in these combinations. In this paper we consider 8 different optimization formulations for computing a single sparse loading vector; these are obtained by combining the following factors: we employ two norms for measuring variance (L2, L1) and two sparsity-inducing norms (L0, L1), which are used in two different ways (constraint, penalty). Three of our formulations, notably the one with L0 constraint and L1 variance, have not been considered in the literature. We give a unifying reformulation which we propose to solve via a natural alternating maximization (AM) method. We show the the AM method is nontrivially equivalent to GPower (Journée et al; JMLR 11:517--553, 2010) for all our formulations. Besides this, we provide 24 efficient parallel SPCA implementations: 3 codes (multi-core, GPU and cluster) for each of the 8 problems. Parallelism in the methods is aimed at i) speeding up computations (our GPU code can be 100 times faster than an efficient serial code written in C++), ii) obtaining solutions explaining more variance and iii) dealing with big data problems (our cluster code is able to solve a 357 GB problem in about a minute).

stat.ML↗

arXiv2020

Efficient Distributed Hessian Free Algorithm for Large-scale Empirical Risk Minimization via Accumulating Sample Strategy

In this paper, we propose a Distributed Accumulated Newton Conjugate gradiEnt (DANCE) method in which sample size is gradually increasing to quickly obtain a solution whose empirical loss is under satisfactory statistical accuracy. Our proposed method is multistage in which the solution of a stage serves as a warm start for the next stage which contains more samples (including the samples in the previous stage). The proposed multistage algorithm reduces the number of passes over data to achieve the statistical accuracy of the full training set. Moreover, our algorithm in nature is easy to be distributed and shares the strong scaling property indicating that acceleration is always expected by using more computing nodes. Various iteration complexity results regarding descent direction computation, communication efficiency and stopping criteria are analyzed under convex setting. Our numerical results illustrate that the proposed method outperforms other comparable methods for solving learning problems including neural networks.

cs.LG↗

arXiv2020

Stochastic Reformulations of Linear Systems: Algorithms and Convergence Theory

We develop a family of reformulations of an arbitrary consistent linear system into a stochastic problem. The reformulations are governed by two user-defined parameters: a positive definite matrix defining a norm, and an arbitrary discrete or continuous distribution over random matrices. Our reformulation has several equivalent interpretations, allowing for researchers from various communities to leverage their domain specific insights. In particular, our reformulation can be equivalently seen as a stochastic optimization problem, stochastic linear system, stochastic fixed point problem and a probabilistic intersection problem. We prove sufficient, and necessary and sufficient conditions for the reformulation to be exact. Further, we propose and analyze three stochastic algorithms for solving the reformulated problem---basic, parallel and accelerated methods---with global linear convergence rates. The rates can be interpreted as condition numbers of a matrix which depends on the system matrix and on the reformulation parameters. This gives rise to a new phenomenon which we call stochastic preconditioning, and which refers to the problem of finding parameters (matrix and distribution) leading to a sufficiently small condition number. Our basic method can be equivalently interpreted as stochastic gradient descent, stochastic Newton method, stochastic proximal point method, stochastic fixed point method, and stochastic projection method, with fixed stepsize (relaxation parameter), applied to the reformulations.

math.NA↗

arXiv2019

Distributed Fixed Point Methods with Compressed Iterates

We propose basic and natural assumptions under which iterative optimization methods with compressed iterates can be analyzed. This problem is motivated by the practice of federated learning, where a large model stored in the cloud is compressed before it is sent to a mobile device, which then proceeds with training based on local data. We develop standard and variance reduced methods, and establish communication complexity bounds. Our algorithms are the first distributed methods with compressed iterates, and the first fixed point methods with compressed iterates.

cs.LG↗

arXiv2019

New Convergence Aspects of Stochastic Gradient Algorithms

The classical convergence analysis of SGD is carried out under the assumption that the norm of the stochastic gradient is uniformly bounded. While this might hold for some loss functions, it is violated for cases where the objective function is strongly convex. In Bottou et al. (2018), a new analysis of convergence of SGD is performed under the assumption that stochastic gradients are bounded with respect to the true gradient norm. We show that for stochastic problems arising in machine learning such bound always holds; and we also propose an alternative convergence analysis of SGD with diminishing learning rate regime. We then move on to the asynchronous parallel setting, and prove convergence of Hogwild! algorithm in the same regime in the case of diminished learning rate. It is well-known that SGD converges if a sequence of learning rates $\{η_t\}$ satisfies $\sum_{t=0}^\infty η_t \rightarrow \infty$ and $\sum_{t=0}^\infty η^2_t < \infty$. We show the convergence of SGD for strongly convex objective function without using bounded gradient assumption when $\{η_t\}$ is a diminishing sequence and $\sum_{t=0}^\infty η_t \rightarrow \infty$. In other words, we extend the current state-of-the-art class of learning rates satisfying the convergence of SGD.

math.OC↗

arXiv2019

FD-Net with Auxiliary Time Steps: Fast Prediction of PDEs using Hessian-Free Trust-Region Methods

Discovering the underlying physical behavior of complex systems is a crucial, but less well-understood topic in many engineering disciplines. This study proposes a finite-difference inspired convolutional neural network framework to learn hidden partial differential equations from given data and iteratively estimate future dynamical behavior. The methodology designs the filter sizes such that they mimic the finite difference between the neighboring points. By learning the governing equation, the network predicts the future evolution of the solution by using only a few trainable parameters. In this paper, we provide numerical results to compare the efficiency of the second-order Trust-Region Conjugate Gradient (TRCG) method with the first-order ADAM optimizer.

cs.LG↗

arXiv2019

A Layered Architecture for Active Perception: Image Classification using Deep Reinforcement Learning

We propose a planning and perception mechanism for a robot (agent), that can only observe the underlying environment partially, in order to solve an image classification problem. A three-layer architecture is suggested that consists of a meta-layer that decides the intermediate goals, an action-layer that selects local actions as the agent navigates towards a goal, and a classification-layer that evaluates the reward and makes a prediction. We design and implement these layers using deep reinforcement learning. A generalized policy gradient algorithm is utilized to learn the parameters of these layers to maximize the expected reward. Our proposed methodology is tested on the MNIST dataset of handwritten digits, which provides us with a level of explainability while interpreting the agent's intermediate goals and course of action.

cs.LG↗

arXiv2019

A Robust Multi-Batch L-BFGS Method for Machine Learning

This paper describes an implementation of the L-BFGS method designed to deal with two adversarial situations. The first occurs in distributed computing environments where some of the computational nodes devoted to the evaluation of the function and gradient are unable to return results on time. A similar challenge occurs in a multi-batch approach in which the data points used to compute function and gradients are purposely changed at each iteration to accelerate the learning process. Difficulties arise because L-BFGS employs gradient differences to update the Hessian approximations, and when these gradients are computed using different data points the updating process can be unstable. This paper shows how to perform stable quasi-Newton updating in the multi-batch setting, studies the convergence properties for both convex and nonconvex functions, and illustrates the behavior of the algorithm in a distributed computing platform on binary classification logistic regression and neural network training problems that arise in machine learning.

math.OC↗

arXiv2019

Multi-Agent Image Classification via Reinforcement Learning

We investigate a classification problem using multiple mobile agents capable of collecting (partial) pose-dependent observations of an unknown environment. The objective is to classify an image over a finite time horizon. We propose a network architecture on how agents should form a local belief, take local actions, and extract relevant features from their raw partial observations. Agents are allowed to exchange information with their neighboring agents to update their own beliefs. It is shown how reinforcement learning techniques can be utilized to achieve decentralized implementation of the classification problem by running a decentralized consensus protocol. Our experimental results on the MNIST handwritten digit dataset demonstrates the effectiveness of our proposed framework.

cs.LG↗

arXiv2019

Don't Forget Your Teacher: A Corrective Reinforcement Learning Framework

Although reinforcement learning (RL) can provide reliable solutions in many settings, practitioners are often wary of the discrepancies between the RL solution and their status quo procedures. Therefore, they may be reluctant to adapt to the novel way of executing tasks proposed by RL. On the other hand, many real-world problems require relatively small adjustments from the status quo policies to achieve improved performance. Therefore, we propose a student-teacher RL mechanism in which the RL (the "student") learns to maximize its reward, subject to a constraint that bounds the difference between the RL policy and the "teacher" policy. The teacher can be another RL policy (e.g., trained under a slightly different setting), the status quo policy, or any other exogenous policy. We formulate this problem using a stochastic optimization model and solve it using a primal-dual policy gradient algorithm. We prove that the policy is asymptotically optimal. However, a naive implementation suffers from high variance and convergence to a stochastic optimal policy. With a few practical adjustments to address these issues, our numerical experiments confirm the effectiveness of our proposed method in multiple GridWorld scenarios.

cs.LG↗

arXiv2018

SGD and Hogwild! Convergence Without the Bounded Gradients Assumption

Stochastic gradient descent (SGD) is the optimization algorithm of choice in many machine learning applications such as regularized empirical risk minimization and training deep neural networks. The classical convergence analysis of SGD is carried out under the assumption that the norm of the stochastic gradient is uniformly bounded. While this might hold for some loss functions, it is always violated for cases where the objective function is strongly convex. In (Bottou et al.,2016), a new analysis of convergence of SGD is performed under the assumption that stochastic gradients are bounded with respect to the true gradient norm. Here we show that for stochastic problems arising in machine learning such bound always holds; and we also propose an alternative convergence analysis of SGD with diminishing learning rate regime, which results in more relaxed conditions than those in (Bottou et al.,2016). We then move on the asynchronous parallel setting, and prove convergence of Hogwild! algorithm in the same regime, obtaining the first convergence results for this method in the case of diminished learning rate.

math.OC↗

arXiv2018

Dual Free Adaptive Mini-batch SDCA for Empirical Risk Minimization

In this paper we develop dual free mini-batch SDCA with adaptive probabilities for regularized empirical risk minimization. This work is motivated by recent work of Shai Shalev-Shwartz on dual free SDCA method, however, we allow a non-uniform selection of "dual" coordinates in SDCA. Moreover, the probability can change over time, making it more efficient than fix uniform or non-uniform selection. We also propose an efficient procedure to generate a random non-uniform mini-batch through iterative process. The work is concluded with multiple numerical experiments to show the efficiency of proposed algorithms.

math.OC↗

arXiv2018

Reinforcement Learning for Solving the Vehicle Routing Problem

We present an end-to-end framework for solving the Vehicle Routing Problem (VRP) using reinforcement learning. In this approach, we train a single model that finds near-optimal solutions for problem instances sampled from a given distribution, only by observing the reward signals and following feasibility rules. Our model represents a parameterized stochastic policy, and by applying a policy gradient algorithm to optimize its parameters, the trained model produces the solution as a sequence of consecutive actions in real time, without the need to re-train for every new problem instance. On capacitated VRP, our approach outperforms classical heuristics and Google's OR-Tools on medium-sized instances in solution quality with comparable computation time (after training). We demonstrate how our approach can handle problems with split delivery and explore the effect of such deliveries on the solution quality. Our proposed framework can be applied to other variants of the VRP such as the stochastic VRP, and has the potential to be applied more generally to combinatorial optimization problems.

cs.AI↗

arXiv2018

Active Metric Learning for Supervised Classification

Clustering and classification critically rely on distance metrics that provide meaningful comparisons between data points. We present mixed-integer optimization approaches to find optimal distance metrics that generalize the Mahalanobis metric extensively studied in the literature. Additionally, we generalize and improve upon leading methods by removing reliance on pre-designated "target neighbors," "triplets," and "similarity pairs." Another salient feature of our method is its ability to enable active learning by recommending precise regions to sample after an optimal metric is computed to improve classification performance. This targeted acquisition can significantly reduce computational burden by ensuring training data completeness, representativeness, and economy. We demonstrate classification and computational performance of the algorithms through several simple and intuitive examples, followed by results on real image and medical datasets.

cs.LG↗

arXiv2018

Stock-out Prediction in Multi-echelon Networks

In multi-echelon inventory systems the performance of a given node is affected by events that occur at many other nodes and in many other time periods. For example, a supply disruption upstream will have an effect on downstream, customer-facing nodes several periods later as the disruption "cascades" through the system. There is very little research on stock-out prediction in single-echelon systems and (to the best of our knowledge) none on multi-echelon systems. However, in real the world, it is clear that there is significant interest in techniques for this sort of stock-out prediction. Therefore, our research aims to fill this gap by using deep neural networks (DNN) to predict stock-outs in multi-echelon supply chains.

cs.LG↗

arXiv2018

Applying Deep Learning to the Newsvendor Problem

The newsvendor problem is one of the most basic and widely applied inventory models. There are numerous extensions of this problem. If the probability distribution of the demand is known, the problem can be solved analytically. However, approximating the probability distribution is not easy and is prone to error; therefore, the resulting solution to the newsvendor problem may be not optimal. To address this issue, we propose an algorithm based on deep learning that optimizes the order quantities for all products based on features of the demand data. Our algorithm integrates the forecasting and inventory-optimization steps, rather than solving them separately, as is typically done, and does not require knowledge of the probability distributions of the demand. Numerical experiments on real-world data suggest that our algorithm outperforms other approaches, including data-driven and machine learning approaches, especially for demands with high volatility. Finally, in order to show how this approach can be used for other inventory optimization problems, we provide an extension for (r,Q) policies.

cs.LG↗

arXiv2017

An Accelerated Communication-Efficient Primal-Dual Optimization Framework for Structured Machine Learning

Distributed optimization algorithms are essential for training machine learning models on very large-scale datasets. However, they often suffer from communication bottlenecks. Confronting this issue, a communication-efficient primal-dual coordinate ascent framework (CoCoA) and its improved variant CoCoA+ have been proposed, achieving a convergence rate of $\mathcal{O}(1/t)$ for solving empirical risk minimization problems with Lipschitz continuous losses. In this paper, an accelerated variant of CoCoA+ is proposed and shown to possess a convergence rate of $\mathcal{O}(1/t^2)$ in terms of reducing suboptimality. The analysis of this rate is also notable in that the convergence rate bounds involve constants that, except in extreme cases, are significantly reduced compared to those previously provided for CoCoA+. The results of numerical experiments are provided to show that acceleration can lead to significant performance gains.

math.OC↗

arXiv2017

SARAH: A Novel Method for Machine Learning Problems Using Stochastic Recursive Gradient

In this paper, we propose a StochAstic Recursive grAdient algoritHm (SARAH), as well as its practical variant SARAH+, as a novel approach to the finite-sum minimization problems. Different from the vanilla SGD and other modern stochastic methods such as SVRG, S2GD, SAG and SAGA, SARAH admits a simple recursive framework for updating stochastic gradient estimates; when comparing to SAG/SAGA, SARAH does not require a storage of past gradients. The linear convergence rate of SARAH is proven under strong convexity assumption. We also prove a linear convergence rate (in the strongly convex case) for an inner loop of SARAH, the property that SVRG does not possess. Numerical experiments demonstrate the efficiency of our algorithm.

stat.ML↗