arXiv subjects

Stephen Boyd

Publications and source records attributed to Stephen Boyd.

At least 109 records · Page 6Linked to original sources

arXiv2019

Learning Convex Optimization Control Policies

Many control policies used in various applications determine the input or action by solving a convex optimization problem that depends on the current state and some parameters. Common examples of such convex optimization control policies (COCPs) include the linear quadratic regulator (LQR), convex model predictive control (MPC), and convex control-Lyapunov or approximate dynamic programming (ADP) policies. These types of control policies are tuned by varying the parameters in the optimization problem, such as the LQR weights, to obtain good performance, judged by application-specific metrics. Tuning is often done by hand, or by simple methods such as a crude grid search. In this paper we propose a method to automate this process, by adjusting the parameters using an approximate gradient of the performance metric with respect to the parameters. Our method relies on recently developed methods that can efficiently evaluate the derivative of the solution of a convex optimization problem with respect to its parameters. We illustrate our method on several examples.

math.OC↗

arXiv2019

A Distributed Method for Fitting Laplacian Regularized Stratified Models

Stratified models are models that depend in an arbitrary way on a set of selected categorical features, and depend linearly on the other features. In a basic and traditional formulation a separate model is fit for each value of the categorical feature, using only the data that has the specific categorical value. To this formulation we add Laplacian regularization, which encourages the model parameters for neighboring categorical values to be similar. Laplacian regularization allows us to specify one or more weighted graphs on the stratification feature values. For example, stratifying over the days of the week, we can specify that the Sunday model parameter should be close to the Saturday and Monday model parameters. The regularization improves the performance of the model over the traditional stratified model, since the model for each value of the categorical `borrows strength' from its neighbors. In particular, it produces a model even for categorical values that did not appear in the training data set. We propose an efficient distributed method for fitting stratified models, based on the alternating direction method of multipliers (ADMM). When the fitting loss functions are convex, the stratified model fitting problem is convex, and our method computes the global minimizer of the loss plus regularization; in other cases it computes a local minimizer. The method is very efficient, and naturally scales to large data sets or numbers of stratified feature values. We illustrate our method with a variety of examples.

cs.LG↗

arXiv2019

Minimizing a Sum of Clipped Convex Functions

We consider the problem of minimizing a sum of clipped convex functions; applications include clipped empirical risk minimization and clipped control. While the problem of minimizing the sum of clipped convex functions is NP-hard, we present some heuristics for approximately solving instances of these problems. These heuristics can be used to find good, if not global, solutions and appear to work well in practice. We also describe an alternative formulation, based on the perspective transformation, which makes the problem amenable to mixed-integer convex programming and yields computationally tractable lower bounds. We illustrate one of our heuristic methods by applying it to various examples and use the perspective transformation to certify that the solutions are relatively close to the global optimum. This paper is accompanied by an open-source implementation.

math.OC↗

arXiv2019

Differentiable Convex Optimization Layers

Recent work has shown how to embed differentiable optimization problems (that is, problems whose solutions can be backpropagated through) as layers within deep learning architectures. This method provides a useful inductive bias for certain problems, but existing software for differentiable optimization layers is rigid and difficult to apply to new settings. In this paper, we propose an approach to differentiating through disciplined convex programs, a subclass of convex optimization problems used by domain-specific languages (DSLs) for convex optimization. We introduce disciplined parametrized programming, a subset of disciplined convex programming, and we show that every disciplined parametrized program can be represented as the composition of an affine map from parameters to problem data, a solver, and an affine map from the solver's solution to a solution of the original problem (a new form we refer to as affine-solver-affine form). We then demonstrate how to efficiently differentiate through each of these components, allowing for end-to-end analytical differentiation through the entire convex program. We implement our methodology in version 1.1 of CVXPY, a popular Python-embedded DSL for convex optimization, and additionally implement differentiable layers for disciplined convex programs in PyTorch and TensorFlow 2.0. Our implementation significantly lowers the barrier to using convex optimization problems in differentiable programs. We present applications in linear machine learning models and in stochastic control, and we show that our layer is competitive (in execution time) compared to specialized differentiable solvers from past work.

cs.LG↗

arXiv2019

Fitting a Kalman Smoother to Data

This paper considers the problem of fitting the parameters of a Kalman smoother to data. We formulate the Kalman smoothing problem with missing measurements as a constrained least squares problem and provide an efficient method to solve it based on sparse linear algebra. We then introduce the Kalman smoother tuning problem, which seeks to find parameters that achieve low prediction error on held out measurements. We derive a Kalman smoother auto-tuning algorithm, which is based on the proximal gradient method, that finds good, if not the best, parameters for a given dataset. Central to our method is the computation of the gradient of the prediction error with respect to the parameters of the Kalman smoother; we describe how to compute this at little to no additional cost. We demonstrate the method on population migration within the United States as well as data collected from an IMU+GPS system while driving. The paper is accompanied by an open-source implementation.

math.OC↗

arXiv2019

Variable Metric Proximal Gradient Method with Diagonal Barzilai-Borwein Stepsize

Variable metric proximal gradient (VM-PG) is a widely used class of convex optimization method. Lately, there has been a lot of research on the theoretical guarantees of VM-PG with different metric selections. However, most such metric selections are dependent on (an expensive) Hessian, or limited to scalar stepsizes like the Barzilai-Borwein (BB) stepsize with lots of safeguarding. Instead, in this paper we propose an adaptive metric selection strategy called the diagonal Barzilai-Borwein (BB) stepsize. The proposed diagonal selection better captures the local geometry of the problem while keeping per-step computation cost similar to the scalar BB stepsize i.e. $O(n)$. Under this metric selection for VM-PG, the theoretical convergence is analyzed. Our empirical studies illustrate the improved convergence results under the proposed diagonal BB stepsize, specifically for ill-conditioned machine learning problems for both synthetic and real-world datasets.

math.OC↗

arXiv2019

Computational Bounds For Photonic Design

Physical design problems, such as photonic inverse design, are typically solved using local optimization methods. These methods often produce what appear to be good or very good designs when compared to classical design methods, but it is not known how far from optimal such designs really are. We address this issue by developing methods for computing a bound on the true optimal value of a physical design problem; physical designs with objective smaller than our bound are impossible to achieve. Our bound is based on Lagrange duality and exploits the special mathematical structure of these physical design problems. For a multi-mode 2D Helmholtz resonator, numerical examples show that the bounds we compute are often close to the objective values obtained using local optimization methods, which reveals that the designs are not only good, but in fact nearly optimal. Our computational bounding method also produces, as a by-product, a reasonable starting point for local optimization methods.

physics.optics↗

arXiv2019

A General Optimization Framework for Dynamic Time Warping

The goal of dynamic time warping is to transform or warp time in order to approximately align two signals together. We pose the choice of warping function as an optimization problem with several terms in the objective. The first term measures the misalignment of the time-warped signals. Two additional regularization terms penalize the cumulative warping and the instantaneous rate of time warping; constraints on the warping can be imposed by assigning the value +inf to the regularization terms. Different choices of the three objective terms yield different time warping functions that trade off signal fit or alignment and properties of the warping function. The optimization problem we formulate is a classical optimal control problem, with initial and terminal constraints, and a state dimension of one. We describe an effective general method that minimizes the objective by discretizing the values of the original and warped time, and using standard dynamic programming to compute the (globally) optimal warping function with the discretized values. Iterated refinement of this scheme yields a high accuracy warping function in just a few iterations. Our method is implemented as an open source Python package GDTW.

cs.LG↗

arXiv2019

Network Optimization for Unified Packet and Circuit Switched Networks

Internet traffic continues to grow relentlessly, driven largely by increasingly high resolution video content. Although studies have shown that the majority of packets processed by Internet routers are pass-through traffic, they nonetheless have to be queued and routed at every hop in current networks, which unnecessarily adds substantial delays and processing costs. Such pass-through traffic can be better circuit-switched through the underlying optical transport network by means of pre-established circuits, which is possible in a unified packet and circuit switched network. In this paper, we propose a novel convex optimization framework based on a new destination-based multicommodity flow formulation for the allocation of circuits in such unified networks. In particular, we consider two deployment settings, one based on real-time traffic monitoring, and the other relying upon history-based traffic predictions. In both cases, we formulate global network optimization objectives as concave functions that capture the fair sharing of network capacity among competing traffic flows. The convexity of our problem formulations ensures globally optimal solutions.

cs.NI↗

arXiv2019

Least Squares Auto-Tuning

Least squares is by far the simplest and most commonly applied computational method in many fields. In almost all applications, the least squares objective is rarely the true objective. We account for this discrepancy by parametrizing the least squares problem and automatically adjusting these parameters using an optimization algorithm. We apply our method, which we call least squares auto-tuning, to data fitting.

math.OC↗

arXiv2019

Disciplined Geometric Programming

We introduce log-log convex programs, which are optimization problems with positive variables that become convex when the variables, objective functions, and constraint functions are replaced with their logs, which we refer to as a log-log transformation. This class of problems generalizes traditional geometric programming and generalized geometric programming, and it includes interesting problems involving nonnegative matrices. We give examples of log-log convex functions, some well-known and some less so, and we develop an analog of disciplined convex programming, which we call disciplined geometric programming. Disciplined geometric programming is a subclass of log-log convex programming generated by a composition rule and a set of functions with known curvature under the log-log transformation. Finally, we describe an implementation of disciplined geometric programming as a reduction in CVXPY 1.0.

math.OC↗

arXiv2019

Dynamic Energy Management

We present a unified method, based on convex optimization, for managing the power produced and consumed by a network of devices over time. We start with the simple setting of optimizing power flows in a static network, and then proceed to the case of optimizing dynamic power flows, i.e., power flows that change with time over a horizon. We leverage this to develop a real-time control strategy, model predictive control, which at each time step solves a dynamic power flow optimization problem, using forecasts of future quantities such as demands, capacities, or prices, to choose the current power flow values. Finally, we consider a useful extension of model predictive control that explicitly accounts for uncertainty in the forecasts. We mirror our framework with an object-oriented software implementation, an open-source Python library for planning and controlling power flows at any scale. We demonstrate our method with various examples. Appendices give more detail about the package, and describe some basic but very effective methods for constructing forecasts from historical data.

math.OC↗

arXiv2019

A Rewriting System for Convex Optimization Problems

We describe a modular rewriting system for translating optimization problems written in a domain-specific language to forms compatible with low-level solver interfaces. Translation is facilitated by reductions, which accept a category of problems and transform instances of that category to equivalent instances of another category. Our system proceeds in two key phases: analysis, in which we attempt to find a suitable solver for a supplied problem, and canonicalization, in which we rewrite the problem in the selected solver's standard form. We implement the described system in version 1.0 of CVXPY, a domain-specific language for mathematical and especially convex optimization. By treating reductions as first-class objects, our method makes it easy to match problems to solvers well-suited for them and to support solvers with a wide variety of standard forms.

math.OC↗

arXiv2018

A Convex Optimization Approach to Radiation Treatment Planning with Dose Constraints

We present a method for handling dose constraints as part of a convex programming framework for inverse treatment planning. Our method uniformly handles mean dose, maximum dose, minimum dose, and dose-volume (i.e., percentile) constraints as part of a convex formulation. Since dose-volume constraints are non-convex, we replace them with a convex restriction. This restriction is, by definition, conservative; to mitigate its impact on the clinical objectives, we develop a two-pass planning algorithm that allows each dose-volume constraint to be met exactly on a second pass by the solver if its corresponding restriction is feasible on the first pass. In another variant, we add slack variables to each dose constraint to prevent the problem from becoming infeasible when the user specifies an incompatible set of constraints or when the constraints are made infeasible by our restriction. Finally, we introduce ConRad, a Python-embedded open-source software package for convex radiation treatment planning. ConRad implements the methods described above and allows users to construct and plan cases through a simple interface.

physics.med-ph↗

arXiv2018

Stochastic Control with Affine Dynamics and Extended Quadratic Costs

An extended quadratic function is a quadratic function plus the indicator function of an affine set, that is, a quadratic function with embedded linear equality constraints. We show that, under some technical conditions, random convex extended quadratic functions are closed under addition, composition with an affine function, expectation, and partial minimization, that is, minimizing over some of its arguments. These properties imply that dynamic programming can be tractably carried out for stochastic control problems with random affine dynamics and extended quadratic cost functions. While the equations for the dynamic programming iterations are much more complicated than for traditional linear quadratic control, they are well suited to an object-oriented implementation, which we describe. We also describe a number of known and new applications.

math.OC↗

arXiv2018

Learning Probabilistic Trajectory Models of Aircraft in Terminal Airspace from Position Data

Models for predicting aircraft motion are an important component of modern aeronautical systems. These models help aircraft plan collision avoidance maneuvers and help conduct offline performance and safety analyses. In this article, we develop a method for learning a probabilistic generative model of aircraft motion in terminal airspace, the controlled airspace surrounding a given airport. The method fits the model based on a historical dataset of radar-based position measurements of aircraft landings and takeoffs at that airport. We find that the model generates realistic trajectories, provides accurate predictions, and captures the statistical properties of aircraft trajectories. Furthermore, the model trains quickly, is compact, and allows for efficient real-time inference.

cs.LG↗

arXiv2018

Globally Convergent Type-I Anderson Acceleration for Non-Smooth Fixed-Point Iterations

We consider the application of the type-I Anderson acceleration to solving general non-smooth fixed-point problems. By interleaving with safe-guarding steps, and employing a Powell-type regularization and a re-start checking for strong linear independence of the updates, we propose the first globally convergent variant of Anderson acceleration assuming only that the fixed-point iteration is non-expansive. We show by extensive numerical experiments that many first order algorithms can be improved, especially in their terminal convergence, with the proposed algorithm. Our proposed method of acceleration is being implemented in SCS 2.0, one of the default solvers used in the convex optimization parser-solver CVXPY 1.0.

math.OC↗

arXiv2018

On the convergence of mirror descent beyond stochastic convex programming

In this paper, we examine the convergence of mirror descent in a class of stochastic optimization problems that are not necessarily convex (or even quasi-convex), and which we call variationally coherent. Since the standard technique of "ergodic averaging" offers no tangible benefits beyond convex programming, we focus directly on the algorithm's last generated sample (its "last iterate"), and we show that it converges with probabiility $1$ if the underlying problem is coherent. We further consider a localized version of variational coherence which ensures local convergence of stochastic mirror descent (SMD) with high probability. These results contribute to the landscape of non-convex stochastic optimization by showing that (quasi-)convexity is not essential for convergence to a global minimum: rather, variational coherence, a much weaker requirement, suffices. Finally, building on the above, we reveal an interesting insight regarding the convergence speed of SMD: in problems with sharp minima (such as generic linear programs or concave minimization problems), SMD reaches a minimum point in a finite number of steps (a.s.), even in the presence of persistent gradient noise. This result is to be contrasted with existing black-box convergence rate estimates that are only asymptotic.

math.OC↗