arXiv subjects

Stephen Boyd

Publications and source records attributed to Stephen Boyd.

At least 73 records · Page 4Linked to original sources

arXiv2022

Optimal Routing for Constant Function Market Makers

We consider the problem of optimally executing an order involving multiple crypto-assets, sometimes called tokens, on a network of multiple constant function market makers (CFMMs). When we ignore the fixed cost associated with executing an order on a CFMM, this optimal routing problem can be cast as a convex optimization problem, which is computationally tractable. When we include the fixed costs, the optimal routing problem is a mixed-integer convex problem, which can be solved using (sometimes slow) global optimization methods, or approximately solved using various heuristics based on convex optimization. The optimal routing problem includes as a special case the problem of identifying an arbitrage present in a network of CFMMs, or certifying that none exists.

math.OC↗

arXiv2022

Embedded Code Generation with CVXPY

We introduce CVXPYgen, a tool for generating custom C code, suitable for embedded applications, that solves a parametrized class of convex optimization problems. CVXPYgen is based on CVXPY, a Python-embedded domain-specific language that supports a natural syntax (that follows the mathematical description) for specifying convex optimization problems. Along with the C implementation of a custom solver, CVXPYgen creates a Python wrapper for prototyping and desktop (non-embedded) applications. We give two examples, position control of a quadcopter and back-testing a portfolio optimization model. CVXPYgen outperforms a state-of-the-art code generation tool in terms of problem size it can handle, binary code size, and solve times. CVXPYgen and the generated solvers are open-source.

math.OC↗

arXiv2021

Minimizing Oracle-Structured Composite Functions

We consider the problem of minimizing a composite convex function with two different access methods: an oracle, for which we can evaluate the value and gradient, and a structured function, which we access only by solving a convex optimization problem. We are motivated by two associated technological developments. For the oracle, systems like PyTorch or TensorFlow can automatically and efficiently compute gradients, given a computation graph description. For the structured function, systems like CVXPY accept a high level domain specific language description of the problem, and automatically translate it to a standard form for efficient solution. We develop a method that makes minimal assumptions about the two functions, does not require the tuning of algorithm parameters, and works well in practice across a variety of problems. Our algorithm combines a number of well-known ideas, including a low-rank quasi-Newton approximation of curvature, piecewise affine lower bounds from bundle-type methods, and two types of damping to ensure stability. We illustrate the method on stochastic optimization, utility maximization, and risk-averse programming problems, showing that our method is more efficient than standard solvers when the oracle function contains much data.

math.OC↗

arXiv2021

Incremental Proximal Multi-Forecast Model Predictive Control

Multi-forecast model predictive control (MF-MPC) is a control policy that creates a plan of actions over a horizon for each of a given set of forecasted scenarios or contingencies, with the constraint that the first action in all plans be the same. In this note we show how these coupled plans can be found by solving a sequence of single plans, using an incremental proximal method. We refer to this policy as incremental proximal model predictive control (IP-MPC). We have observed that even when the iterations in IP-MPC are not carried out to convergence, we obtain a policy that achieves much of the improvement of MF-MPC over single-forecast model predictive control (MPC).

math.OC↗

arXiv2021

Solving Large-Scale Granular Resource Allocation Problems Efficiently with POP

Resource allocation problems in many computer systems can be formulated as mathematical optimization problems. However, finding exact solutions to these problems using off-the-shelf solvers is often intractable for large problem sizes with tight SLAs, leading system designers to rely on cheap, heuristic algorithms. We observe, however, that many allocation problems are granular: they consist of a large number of clients and resources, each client requests a small fraction of the total number of resources, and clients can interchangeably use different resources. For these problems, we propose an alternative approach that reuses the original optimization problem formulation and leads to better allocations than domain-specific heuristics. Our technique, Partitioned Optimization Problems (POP), randomly splits the problem into smaller problems (with a subset of the clients and resources in the system) and coalesces the resulting sub-allocations into a global allocation for all clients. We provide theoretical and empirical evidence as to why random partitioning works well. In our experiments, POP achieves allocations within 1.5% of the optimal with orders-of-magnitude improvements in runtime compared to existing systems for cluster scheduling, traffic engineering, and load balancing.

cs.DC↗

arXiv2021

Minimum-Distortion Embedding

We consider the vector embedding problem. We are given a finite set of items, with the goal of assigning a representative vector to each one, possibly under some constraints (such as the collection of vectors being standardized, i.e., having zero mean and unit covariance). We are given data indicating that some pairs of items are similar, and optionally, some other pairs are dissimilar. For pairs of similar items, we want the corresponding vectors to be near each other, and for dissimilar pairs, we want the corresponding vectors to not be near each other, measured in Euclidean distance. We formalize this by introducing distortion functions, defined for some pairs of the items. Our goal is to choose an embedding that minimizes the total distortion, subject to the constraints. We call this the minimum-distortion embedding (MDE) problem. The MDE framework is simple but general. It includes a wide variety of embedding methods, such as spectral embedding, principal component analysis, multidimensional scaling, dimensionality reduction methods (like Isomap and UMAP), force-directed layout, and others. It also includes new embeddings, and provides principled ways of validating historical and new embeddings alike. We develop a projected quasi-Newton method that approximately solves MDE problems and scales to large data sets. We implement this method in PyMDE, an open-source Python package. In PyMDE, users can select from a library of distortion functions and constraints or specify custom ones, making it easy to rapidly experiment with different embeddings. Our software scales to data sets with millions of items and tens of millions of distortion functions. To demonstrate our method, we compute embeddings for several real-world data sets, including images, an academic co-author network, US county demographic data, and single-cell mRNA transcriptomes.

cs.LG↗

arXiv2021

Constant Function Market Makers: Multi-Asset Trades via Convex Optimization

The rise of Ethereum and other blockchains that support smart contracts has led to the creation of decentralized exchanges (DEXs), such as Uniswap, Balancer, Curve, mStable, and SushiSwap, which enable agents to trade cryptocurrencies without trusting a centralized authority. While traditional exchanges use order books to match and execute trades, DEXs are typically organized as constant function market makers (CFMMs). CFMMs accept and reject proposed trades based on the evaluation of a function that depends on the proposed trade and the current reserves of the DEX. For trades that involve only two assets, CFMMs are easy to understand, via two functions that give the quantity of one asset that must be tendered to receive a given quantity of the other, and vice versa. When more than two assets are being exchanged, it is harder to understand the landscape of possible trades. We observe that various problems of choosing a multi-asset trade can be formulated as convex optimization problems, and can therefore be reliably and efficiently solved.

math.OC↗

arXiv2021

Distributional Robust Kelly Gambling: Optimal Strategy under Uncertainty in the Long-Run

In classic Kelly gambling, bets are chosen to maximize the expected log growth of wealth, under a known probability distribution. Breiman provides rigorous mathematical proofs that Kelly strategy maximizes the rate of asset growth (asymptotically maximal magnitude property), which is thought of as the principal justification for selecting expected logarithmic utility as the guide to portfolio selection. Despite very nice theoretical properties, the classic Kelly strategy is rarely used in practical portfolio allocation directly due to practically unavoidable uncertainty. In this paper we consider the distributional robust version of the Kelly gambling problem, in which the probability distribution is not known, but lies in a given set of possible distributions. The bet is chosen to maximize the worst-case (smallest) expected log growth among the distributions in the given set. Computationally, this distributional robust Kelly gambling problem is convex, but in general need not be tractable. We show that it can be tractably solved in a number of useful cases when there is a finite number of outcomes with standard tools from disciplined convex programming. Theoretically, in sequential decision making with varying distribution within a given uncertainty set, we prove that distributional robust Kelly strategy asymptotically maximizes the worst-case rate of asset growth, and dominants any other essentially different strategy by magnitude. Our results extends Breiman's theoretical result and justifies that the distributional robust Kelly strategy is the optimal strategy in the long-run for practical betting with uncertainty.

math.OC↗

arXiv2021

Optimal Claiming of Social Security Benefits

Using a lifecycle framework with Epstein-Zin (1989) utility and a mixed-integer optimization approach, we compute the optimal age to claim Social Security benefits. Taking advantage of homogeneity, a sufficient statistic is the ratio of wealth to the primary insurance amount (PIA). If the investor's wealth to PIA ratio exceeds a certain threshold, individuals should defer Social Security for at least a year. The optimal threshold depends on mortality assumptions and an individual's utility preferences, but is less sensitive to capital market assumptions. The threshold wealth to PIA ratio increases from 5.5 for men and 5.2 for women at age 62 to 11.1 for men and 10.4 for women at age 69. Below the threshold wealth to PIA ratio, individuals claim Social Security to raise consumption. Above this level, investors can afford to fund consumption out of wealth for at least one year, and then claim a higher benefit.

physics.soc-ph↗

arXiv2021

Dirty Pixels: Towards End-to-End Image Processing and Perception

Real-world imaging systems acquire measurements that are degraded by noise, optical aberrations, and other imperfections that make image processing for human viewing and higher-level perception tasks challenging. Conventional cameras address this problem by compartmentalizing imaging from high-level task processing. As such, conventional imaging involves processing the RAW sensor measurements in a sequential pipeline of steps, such as demosaicking, denoising, deblurring, tone-mapping and compression. This pipeline is optimized to obtain a visually pleasing image. High-level processing, on the other hand, involves steps such as feature extraction, classification, tracking, and fusion. While this siloed design approach allows for efficient development, it also dictates compartmentalized performance metrics, without knowledge of the higher-level task of the camera system. For example, today's demosaicking and denoising algorithms are designed using perceptual image quality metrics but not with domain-specific tasks such as object detection in mind. We propose an end-to-end differentiable architecture that jointly performs demosaicking, denoising, deblurring, tone-mapping, and classification. The architecture learns processing pipelines whose outputs differ from those of existing ISPs optimized for perceptual quality, preserving fine detail at the cost of increased noise and artifacts. We demonstrate on captured and simulated data that our model substantially improves perception in low light and other challenging conditions, which is imperative for real-world applications. Finally, we found that the proposed model also achieves state-of-the-art accuracy when optimized for image reconstruction in low-light conditions, validating the architecture itself as a potentially useful drop-in network for reconstruction and analysis tasks beyond the applications demonstrated in this work.

cs.CV↗

arXiv2021

Allocation of Fungible Resources via a Fast, Scalable Price Discovery Method

We consider the problem of assigning or allocating resources to a set of jobs. We consider the case when the resources are fungible, that is, the job can be done with any mix of the resources, but with different efficiencies. In our formulation we maximize a total utility subject to a given limit on the resource usage, which is a convex optimization problem and so is tractable. In this paper we develop a custom, parallelizable algorithm for solving the resource allocation problem that scales to large problems, with millions of jobs. Our algorithm is based on the dual problem, in which the dual variables associated with the resource usage limit can be interpreted as resource prices. Our method updates the resource prices in each iteration, ultimately discovering the optimal resource prices, from which an optimal allocation is obtained. We provide an open-source implementation of our method, which can solve problems with millions of jobs in a few seconds on CPU, and under a second on a GPU; our software can solve smaller problems in milliseconds. On large problems, our implementation is up to three orders of magnitude faster than a commerical solver for convex optimization.

math.OC↗

arXiv2021

Tax-Aware Portfolio Construction via Convex Optimization

We describe an optimization-based tax-aware portfolio construction method that adds tax liability to standard Markowitz-based portfolio construction. Our method produces a trade list that specifies the number of shares to buy of each asset and the number of shares to sell from each tax lot held. To avoid wash sales (in which some realized capital losses are disallowed), we assume that we trade monthly, and cannot simultaneously buy and sell the same asset. The tax-aware portfolio construction problem is not convex, but it becomes convex when we specify, for each asset, whether we buy or sell it. It can be solved using standard mixed-integer convex optimization methods at the cost of very long solve times for some problem instances. We present a custom convex relaxation of the problem that borrows curvature from the risk model. This relaxation can provide a good approximation of the true tax liability, while greatly enhancing computational tractability. This method requires the solution of only two convex optimization problems: the first determines whether we buy or sell each asset, and the second generates the final trade list. In our numerical experiments, our method almost always solves the nonconvex problem to optimality, and when it does not, it produces a trade list very close to optimal. Backtests show that the performance of our method is indistinguishable from that obtained using a globally optimal solution, but with significantly reduced computational effort.

math.OC↗

arXiv2021

Portfolio Performance Attribution via Shapley Value

We consider an investment process that includes a number of features, each of which can be active or inactive. Our goal is to attribute or decompose an achieved performance to each of these features, plus a baseline value. There are many ways to do this, which lead to potentially different attributions in any specific case. We argue that a specific attribution method due to Shapley is the preferred method, and discuss methods that can be used to compute this attribution exactly, or when that is not practical, approximately.

q-fin.CP↗

arXiv2021

Portfolio Construction Using Stratified Models

In this paper we develop models of asset return mean and covariance that depend on some observable market conditions, and use these to construct a trading policy that depends on these conditions, and the current portfolio holdings. After discretizing the market conditions, we fit Laplacian regularized stratified models for the return mean and covariance. These models have a different mean and covariance for each market condition, but are regularized so that nearby market conditions have similar models. This technique allows us to fit models for market conditions that have not occurred in the training data, by borrowing strength from nearby market conditions for which we do have data. These models are combined with a Markowitz-inspired optimization method to yield a trading policy that is based on market conditions. We illustrate our method on a small universe of 18 ETFs, using three well known and publicly available market variables to construct 1000 market conditions, and show that it performs well out of sample. The method, however, is general, and scales to much larger problems, that presumably would use proprietary data sources and forecasts along with publicly available data.

q-fin.PM↗

arXiv2021

Pareto Optimal Projection Search (POPS): Automated Radiation Therapy Treatment Planning by Direct Search of the Pareto Surface

Objective: Radiation therapy treatment planning is a time-consuming, iterative process with potentially high inter-planner variability. Fully automated treatment planning processes could reduce a planner's active treatment planning time and remove inter-planner variability, with the potential to tremendously improve patient turnover and quality of care. In developing fully automated algorithms for treatment planning, we have two main objectives: to produce plans that are 1) Pareto optimal and 2) clinically acceptable. Here, we propose the Pareto optimal projection search (POPS) algorithm, which provides a general framework for directly searching the Pareto front. Methods: Our POPS algorithm is a novel automated planning method that combines two main search processes: 1) gradient-free search in the decision variable space and 2) projection of decision variables to the Pareto front using the bisection method. We demonstrate the performance of POPS by comparing with clinical treatment plans. As one possible quantitative measure of treatment plan quality, we construct a clinical acceptability scoring function (SF) modified from the previously developed general evaluation metric (GEM). Results: On a dataset of 21 prostate cases collected as part of clinical workflow, our proposed POPS algorithm produces Pareto optimal plans that are clinically acceptable in regards to dose conformity, dose homogeneity, and sparing of organs-at-risk. Conclusion: Our proposed POPS algorithm provides a general framework for fully automated treatment planning that achieves clinically acceptable dosimetric quality without requiring active planning from human planners. Significance: Our fully automated POPS algorithm addresses many key limitations of other automated planning approaches, and we anticipate that it will substantially improve treatment planning workflow.

physics.med-ph↗

arXiv2021

Low Rank Forecasting

We consider the problem of forecasting multiple values of the future of a vector time series, using some past values. This problem, and related ones such as one-step-ahead prediction, have a very long history, and there are a number of well-known methods for it, including vector auto-regressive models, state-space methods, multi-task regression, and others. Our focus is on low rank forecasters, which break forecasting up into two steps: estimating a vector that can be interpreted as a latent state, given the past, and then estimating the future values of the time series, given the latent state estimate. We introduce the concept of forecast consistency, which means that the estimates of the same value made at different times are consistent. We formulate the forecasting problem in general form, and focus on linear forecasters, for which we propose a formulation that can be solved via convex optimization. We describe a number of extensions and variations, including nonlinear forecasters, data weighting, the inclusion of auxiliary data, and additional objective terms. We illustrate our methods with several examples.

stat.ML↗

arXiv2021

Covariance Prediction via Convex Optimization

We consider the problem of predicting the covariance of a zero mean Gaussian vector, based on another feature vector. We describe a covariance predictor that has the form of a generalized linear model, i.e., an affine function of the features followed by an inverse link function that maps vectors to symmetric positive definite matrices. The log-likelihood is a concave function of the predictor parameters, so fitting the predictor involves convex optimization. Such predictors can be combined with others, or recursively applied to improve performance.

stat.ML↗

arXiv2021

A Certainty Equivalent Merton Problem

The Merton problem is the well-known stochastic control problem of choosing consumption over time, as well as an investment mix, to maximize expected constant relative risk aversion (CRRA) utility of consumption. Merton formulated the problem and provided an analytical solution in 1970; since then a number of extensions of the original formulation have been solved. In this note we identify a certainty equivalent problem, i.e., a deterministic optimal control problem with the same optimal value function and optimal policy, for the base Merton problem, as well as a number of extensions. When time is discretized, the certainty equivalent problem becomes a second-order cone program (SOCP), readily formulated and solved using domain specific languages for convex optimization. This makes it a good starting point for model predictive control, a policy that can handle extensions that are either too cumbersome or impossible to handle exactly using standard dynamic programming methods.

math.OC↗