Tradeoffs between Convergence Speed and Reconstruction Accuracy in Inverse Problems

Solving inverse problems with iterative algorithms is popular, especially for large data. Due to time constraints, the number of possible iterations is usually limited, potentially affecting the achievable accuracy. Given an error one is willing to tolerate, an important question is whether it is possible to modify the original iterations to obtain faster convergence to a minimizer achieving the allowed error without increasing the computational cost of each iteration considerably. Relying on recent recovery techniques developed for settings in which the desired signal belongs to some low-dimensional set, we show that using a coarse estimate of this set may lead to faster convergence at the cost of an additional reconstruction error related to the accuracy of the set approximation. Our theory ties to recent advances in sparse recovery, compressed sensing, and deep learning. Particularly, it may provide a possible explanation to the successful approximation of the l1-minimization solution by neural networks with layers representing iterations, as practiced in the learned iterative shrinkage-thresholding algorithm (LISTA).Comment: To appear in IEEE Transactions on Signal Processin

Efficiently testing local optimality and escaping saddles for ReLU networks

We provide a theoretical algorithm for checking local optimality and escaping saddles at nondifferentiable points of empirical risks of two-layer ReLU networks. Our algorithm receives any parameter value and returns: local minimum, second-order stationary point, or a strict descent direction. The presence of $M$ data points on the nondifferentiability of the ReLU divides the parameter space into at most $2^M$ regions, which makes analysis difficult. By exploiting polyhedral geometry, we reduce the total computation down to one convex quadratic program (QP) for each hidden node, $O(M)$ (in)equality tests, and one (or a few) nonconvex QP. For the last QP, we show that our specific problem can be solved efficiently, in spite of nonconvexity. In the benign case, we solve one equality constrained QP, and we prove that projected gradient descent solves it exponentially fast. In the bad case, we have to solve a few more inequality constrained QPs, but we prove that the time complexity is exponential only in the number of inequality constraints. Our experiments show that either benign case or bad case with very few inequality constraints occurs, implying that our algorithm is efficient in most cases.Comment: 23 pages, appeared at ICLR 201

Quadratically constrained quadratic programming for classification using particle swarms and applications

Particle swarm optimization is used in several combinatorial optimization problems. In this work, particle swarms are used to solve quadratic programming problems with quadratic constraints. The approach of particle swarms is an example for interior point methods in optimization as an iterative technique. This approach is novel and deals with classification problems without the use of a traditional classifier. Our method determines the optimal hyperplane or classification boundary for a data set. In a binary classification problem, we constrain each class as a cluster, which is enclosed by an ellipsoid. The estimation of the optimal hyperplane between the two clusters is posed as a quadratically constrained quadratic problem. The optimization problem is solved in distributed format using modified particle swarms. Our method has the advantage of using the direction towards optimal solution rather than searching the entire feasible region. Our results on the Iris, Pima, Wine, and Thyroid datasets show that the proposed method works better than a neural network and the performance is close to that of SVM.Comment: 17 pages, 3 figure

Robust Dynamic Locomotion via Reinforcement Learning and Novel Whole Body Controller

We propose a robust dynamic walking controller consisting of a dynamic locomotion planner, a reinforcement learning process for robustness, and a novel whole-body locomotion controller (WBLC). Previous approaches specify either the position or the timing of steps, however, the proposed locomotion planner simultaneously computes both of these parameters as locomotion outputs. Our locomotion strategy relies on devising a reinforcement learning (RL) approach for robust walking. The learned policy generates multi step walking patterns, and the process is quick enough to be suitable for real-time controls. For learning, we devise an RL strategy that uses a phase space planner (PSP) and a linear inverted pendulum model to make the problem tractable and very fast. Then, the learned policy is used to provide goal-based commands to the WBLC, which calculates the torque commands to be executed in full-humanoid robots. The WBLC combines multiple prioritized tasks and calculates the associated reaction forces based on practical inequality constraints. The novel formulation includes efficient calculation of the time derivatives of various Jacobians. This provides high-fidelity dynamic control of fast motions. More specifically, we compute the time derivative of the Jacobian for various tasks and the Jacobian of the centroidal momentum task by utilizing Lie group operators and operational space dynamics respectively. The integration of RL-PSP and the WBLC provides highly robust, versatile, and practical locomotion including steering while walking and handling push disturbances of up to 520 N during an interval of 0.1 sec. Theoretical and numerical results are tested through a 3D physics-based simulation of the humanoid robot Valkyrie.Comment: 15 pages, 12 figure

Initialization-free Distributed Algorithms for Optimal Resource Allocation with Feasibility Constraints and its Application to Economic Dispatch of Power Systems

In this paper, the distributed resource allocation optimization problem is investigated. The allocation decisions are made to minimize the sum of all the agents' local objective functions while satisfying both the global network resource constraint and the local allocation feasibility constraints. Here the data corresponding to each agent in this separable optimization problem, such as the network resources, the local allocation feasibility constraint, and the local objective function, is only accessible to individual agent and cannot be shared with others, which renders new challenges in this distributed optimization problem. Based on either projection or differentiated projection, two classes of continuous-time algorithms are proposed to solve this distributed optimization problem in an initialization-free and scalable manner. Thus, no re-initialization is required even if the operation environment or network configuration is changed, making it possible to achieve a "plug-and-play" optimal operation of networked heterogeneous agents. The algorithm convergence is guaranteed for strictly convex objective functions, and the exponential convergence is proved for strongly convex functions without local constraints. Then the proposed algorithm is applied to the distributed economic dispatch problem in power grids, to demonstrate how it can achieve the global optimum in a scalable way, even when the generation cost, or system load, or network configuration, is changing.Comment: 13 pages, 7 figure

Distributed coordination for nonsmooth convex optimization via saddle-point dynamics

This paper considers continuous-time coordination algorithms for networks of agents that seek to collectively solve a general class of nonsmooth convex optimization problems with an inherent distributed structure. Our algorithm design builds on the characterization of the solutions of the nonsmooth convex program as saddle points of an augmented Lagrangian. We show that the associated saddle-point dynamics are asymptotically correct but, in general, not distributed because of the presence of a global penalty parameter. This motivates the design of a discontinuous saddle-point-like algorithm that enjoys the same convergence properties and is fully amenable to distributed implementation. Our convergence proofs rely on the identification of a novel global Lyapunov function for saddle-point dynamics. This novelty also allows us to identify mild convexity and regularity conditions on the objective function that guarantee the exponential convergence rate of the proposed algorithms for convex optimization problems subject to equality constraints. Various examples illustrate our discussion.Comment: 20 page

Proximal Reinforcement Learning: A New Theory of Sequential Decision Making in Primal-Dual Spaces

In this paper, we set forth a new vision of reinforcement learning developed by us over the past few years, one that yields mathematically rigorous solutions to longstanding important questions that have remained unresolved: (i) how to design reliable, convergent, and robust reinforcement learning algorithms (ii) how to guarantee that reinforcement learning satisfies pre-specified "safety" guarantees, and remains in a stable region of the parameter space (iii) how to design "off-policy" temporal difference learning algorithms in a reliable and stable manner, and finally (iv) how to integrate the study of reinforcement learning into the rich theory of stochastic optimization. In this paper, we provide detailed answers to all these questions using the powerful framework of proximal operators. The key idea that emerges is the use of primal dual spaces connected through the use of a Legendre transform. This allows temporal difference updates to occur in dual spaces, allowing a variety of important technical advantages. The Legendre transform elegantly generalizes past algorithms for solving reinforcement learning problems, such as natural gradient methods, which we show relate closely to the previously unconnected framework of mirror descent methods. Equally importantly, proximal operator theory enables the systematic development of operator splitting methods that show how to safely and reliably decompose complex products of gradients that occur in recent variants of gradient-based temporal difference learning. This key technical innovation makes it possible to finally design "true" stochastic gradient methods for reinforcement learning. Finally, Legendre transforms enable a variety of other benefits, including modeling sparsity and domain geometry. Our work builds extensively on recent work on the convergence of saddle-point algorithms, and on the theory of monotone operators.Comment: 121 page

Differentiating through Log-Log Convex Programs

We show how to efficiently compute the derivative (when it exists) of the solution map of log-log convex programs (LLCPs). These are nonconvex, nonsmooth optimization problems with positive variables that become convex when the variables, objective functions, and constraint functions are replaced with their logs. We focus specifically on LLCPs generated by disciplined geometric programming, a grammar consisting of a set of atomic functions with known log-log curvature and a composition rule for combining them. We represent a parametrized LLCP as the composition of a smooth transformation of parameters, a convex optimization problem, and an exponential transformation of the convex optimization problem's solution. The derivative of this composition can be computed efficiently, using recently developed methods for differentiating through convex optimization problems. We implement our method in CVXPY, a Python-embedded modeling language and rewriting system for convex optimization. In just a few lines of code, a user can specify a parametrized LLCP, solve it, and evaluate the derivative or its adjoint at a vector. This makes it possible to conduct sensitivity analyses of solutions, given perturbations to the parameters, and to compute the gradient of a function of the solution with respect to the parameters. We use the adjoint of the derivative to implement differentiable log-log convex optimization layers in PyTorch and TensorFlow. Finally, we present applications to designing queuing systems and fitting structured prediction models.Comment: Fix some typo

Level-set methods for convex optimization

Convex optimization problems arising in applications often have favorable objective functions and complicated constraints, thereby precluding first-order methods from being immediately applicable. We describe an approach that exchanges the roles of the objective and constraint functions, and instead approximately solves a sequence of parametric level-set problems. A zero-finding procedure, based on inexact function evaluations and possibly inexact derivative information, leads to an efficient solution scheme for the original problem. We describe the theoretical and practical properties of this approach for a broad range of problems, including low-rank semidefinite optimization, sparse optimization, and generalized linear models for inference.Comment: 38 page

A Tropical Approach to Neural Networks with Piecewise Linear Activations

We present a new, unifying approach following some recent developments on the complexity of neural networks with piecewise linear activations. We treat neural network layers with piecewise linear activations as tropical polynomials, which generalize polynomials in the so-called $(\max, +)$ or tropical algebra, with possibly real-valued exponents. Motivated by the discussion in (arXiv:1402.1869), this approach enables us to refine their upper bounds on linear regions of layers with ReLU or leaky ReLU activations to $\min\left\{ 2^m, \sum_{j=0}^n \binom{m}{j} \right\}$, where $n, m$ are the number of inputs and outputs, respectively. Additionally, we recover their upper bounds on maxout layers. Our work follows a novel path, exclusively under the lens of tropical geometry, which is independent of the improvements reported in (arXiv:1611.01491, arXiv:1711.02114). Finally, we present a geometric approach for effective counting of linear regions using random sampling in order to avoid the computational overhead of exact counting approachesComment: v2: Removed morphological perceptron section and added vertex sampling section. Updated references. 18 pages, 7 figure