skip to main content

Title: Efficient Identification of Error-in-Variables Switched Systems via a Sum-of-Squares Polynomial Based Subspace Clustering Method
This paper addresses the problem of identification of error in variables switched linear models from experimental input/output data. This problem is known to be generically NP hard and thus computationally expensive to solve. To address this difficulty, several relaxations have been proposed in the past few years. While solvable in polynomial time these (convex) relaxations tend to scale poorly with the number of points and number/order of the subsystems, effectively limiting their applicability to scenarios with relatively small number of data points. To address this difficulty, in this paper we propose an efficient method that only requires performing (number of subsystems) singular value decompositions of matrices whose size is independent of the number of points. The underlying idea is to obtain a sum-of-squares polynomial approximation of the support of each subsystem one-at-a-time, and use these polynomials to segment the data into sets, each generated by a single subsystem. As shown in the paper, exploiting ideas from Christoffel's functions allows for finding these polynomial approximations simply by performing SVDs. The parameters of each subsystem can then be identified from the segmented data using existing error-in-variables (EIV) techniques.
Authors:
; ;
Award ID(s):
1638234 1814631 1646121
Publication Date:
NSF-PAR ID:
10176079
Journal Name:
2019 IEEE 58th Conference on Decision and Control (CDC)
Page Range or eLocation-ID:
3429 to 3434
Sponsoring Org:
National Science Foundation
More Like this
  1. Obeid, I. ; Selesnik, I. ; Picone, J. (Ed.)
    The Neuronix high-performance computing cluster allows us to conduct extensive machine learning experiments on big data [1]. This heterogeneous cluster uses innovative scheduling technology, Slurm [2], that manages a network of CPUs and graphics processing units (GPUs). The GPU farm consists of a variety of processors ranging from low-end consumer grade devices such as the Nvidia GTX 970 to higher-end devices such as the GeForce RTX 2080. These GPUs are essential to our research since they allow extremely compute-intensive deep learning tasks to be executed on massive data resources such as the TUH EEG Corpus [2]. We use TensorFlow [3] as the core machine learning library for our deep learning systems, and routinely employ multiple GPUs to accelerate the training process. Reproducible results are essential to machine learning research. Reproducibility in this context means the ability to replicate an existing experiment – performance metrics such as error rates should be identical and floating-point calculations should match closely. Three examples of ways we typically expect an experiment to be replicable are: (1) The same job run on the same processor should produce the same results each time it is run. (2) A job run on a CPU and GPU should producemore »identical results. (3) A job should produce comparable results if the data is presented in a different order. System optimization requires an ability to directly compare error rates for algorithms evaluated under comparable operating conditions. However, it is a difficult task to exactly reproduce the results for large, complex deep learning systems that often require more than a trillion calculations per experiment [5]. This is a fairly well-known issue and one we will explore in this poster. Researchers must be able to replicate results on a specific data set to establish the integrity of an implementation. They can then use that implementation as a baseline for comparison purposes. A lack of reproducibility makes it very difficult to debug algorithms and validate changes to the system. Equally important, since many results in deep learning research are dependent on the order in which the system is exposed to the data, the specific processors used, and even the order in which those processors are accessed, it becomes a challenging problem to compare two algorithms since each system must be individually optimized for a specific data set or processor. This is extremely time-consuming for algorithm research in which a single run often taxes a computing environment to its limits. Well-known techniques such as cross-validation [5,6] can be used to mitigate these effects, but this is also computationally expensive. These issues are further compounded by the fact that most deep learning algorithms are susceptible to the way computational noise propagates through the system. GPUs are particularly notorious for this because, in a clustered environment, it becomes more difficult to control which processors are used at various points in time. Another equally frustrating issue is that upgrades to the deep learning package, such as the transition from TensorFlow v1.9 to v1.13, can also result in large fluctuations in error rates when re-running the same experiment. Since TensorFlow is constantly updating functions to support GPU use, maintaining an historical archive of experimental results that can be used to calibrate algorithm research is quite a challenge. This makes it very difficult to optimize the system or select the best configurations. The overall impact of all of these issues described above is significant as error rates can fluctuate by as much as 25% due to these types of computational issues. Cross-validation is one technique used to mitigate this, but that is expensive since you need to do multiple runs over the data, which further taxes a computing infrastructure already running at max capacity. GPUs are preferred when training a large network since these systems train at least two orders of magnitude faster than CPUs [7]. Large-scale experiments are simply not feasible without using GPUs. However, there is a tradeoff to gain this performance. Since all our GPUs use the NVIDIA CUDA® Deep Neural Network library (cuDNN) [8], a GPU-accelerated library of primitives for deep neural networks, it adds an element of randomness into the experiment. When a GPU is used to train a network in TensorFlow, it automatically searches for a cuDNN implementation. NVIDIA’s cuDNN implementation provides algorithms that increase the performance and help the model train quicker, but they are non-deterministic algorithms [9,10]. Since our networks have many complex layers, there is no easy way to avoid this randomness. Instead of comparing each epoch, we compare the average performance of the experiment because it gives us a hint of how our model is performing per experiment, and if the changes we make are efficient. In this poster, we will discuss a variety of issues related to reproducibility and introduce ways we mitigate these effects. For example, TensorFlow uses a random number generator (RNG) which is not seeded by default. TensorFlow determines the initialization point and how certain functions execute using the RNG. The solution for this is seeding all the necessary components before training the model. This forces TensorFlow to use the same initialization point and sets how certain layers work (e.g., dropout layers). However, seeding all the RNGs will not guarantee a controlled experiment. Other variables can affect the outcome of the experiment such as training using GPUs, allowing multi-threading on CPUs, using certain layers, etc. To mitigate our problems with reproducibility, we first make sure that the data is processed in the same order during training. Therefore, we save the data from the last experiment and to make sure the newer experiment follows the same order. If we allow the data to be shuffled, it can affect the performance due to how the model was exposed to the data. We also specify the float data type to be 32-bit since Python defaults to 64-bit. We try to avoid using 64-bit precision because the numbers produced by a GPU can vary significantly depending on the GPU architecture [11-13]. Controlling precision somewhat reduces differences due to computational noise even though technically it increases the amount of computational noise. We are currently developing more advanced techniques for preserving the efficiency of our training process while also maintaining the ability to reproduce models. In our poster presentation we will demonstrate these issues using some novel visualization tools, present several examples of the extent to which these issues influence research results on electroencephalography (EEG) and digital pathology experiments and introduce new ways to manage such computational issues.« less
  2. We present a principal-agent model of a one-shot, shallow, systems engineering process. The process is "one-shot" in the sense that decisions are made during a one-time step and that they are final. The term "shallow" refers to a one-layer hierarchy of the process. Specifically, we assume that the systems engineer has already decomposed the problem in subsystems and that each subsystem is assigned to a different subsystem engineer. Each subsystem engineer works independently to maximize their own expected payoff. The goal of the systems engineer is to maximize the system-level payoff by incentivizing the subsystem engineers. We restrict our attention to requirements-based system-level payoffs, i.e., the systems engineer makes a profit only if all the design requirements are met. We illustrate the model using the design of an Earth-orbiting satellite system where the systems engineer determines the optimum incentive structures and requirements for two subsystems: the propulsion subsystem and the power subsystem. The model enables the analysis of a systems engineer's decisions about optimal passed-down requirements and incentives for sub-system engineers under different levels of task difficulty and associated costs. Sample results, for the case of risk-neutral systems and subsystems engineers, show that it is not always in the bestmore »interest of the systems engineer to pass down the true requirements. As expected, the model predicts that for small to moderate task uncertainties the optimal requirements are higher than the true ones, effectively eliminating the probability of failure for the systems engineer. In contrast, the model predicts that for large task uncertainties the optimal requirements should be smaller than the true ones in order to lure the subsystem engineers into participation.« less
  3. In this paper, we first introduce a variational formulation of the Unit Commitment (UC) problem, in which generation and ramping trajectories of the generating units are continuous time signals and the generating units cost depends on the three signals: the binary commitment status of the units as well as their continuous-time generation and ramping trajectories. We assume such bids are piecewise strictly convex time-varying linear functions of these three variables. Based on this problem derive a tractable approximation by constraining the commitment trajectories to switch in a discrete and finite set of points and representing the trajectories in the function space of piece-wise polynomial functions within the intervals, whose discrete coefficients are then the UC problem decision variables. Our judicious choice of the signal space allows us to represent cost and constraints as linear functions of such coefficients, thus, our UC models preserves the MILP formulation of the UC problem. Numerical simulation over real load data from the California ISO demonstrate that the proposed UC model reduces the total dayahead and real-time operation cost, and the number of ramping scarcity events in the real-time operations.
  4. Abstract Estimating the mean of a probability distribution using i.i.d. samples is a classical problem in statistics, wherein finite-sample optimal estimators are sought under various distributional assumptions. In this paper, we consider the problem of mean estimation when independent samples are drawn from $d$-dimensional non-identical distributions possessing a common mean. When the distributions are radially symmetric and unimodal, we propose a novel estimator, which is a hybrid of the modal interval, shorth and median estimators and whose performance adapts to the level of heterogeneity in the data. We show that our estimator is near optimal when data are i.i.d. and when the fraction of ‘low-noise’ distributions is as small as $\varOmega \left (\frac{d \log n}{n}\right )$, where $n$ is the number of samples. We also derive minimax lower bounds on the expected error of any estimator that is agnostic to the scales of individual data points. Finally, we extend our theory to linear regression. In both the mean estimation and regression settings, we present computationally feasible versions of our estimators that run in time polynomial in the number of data points.
  5. We present approximation and exact algorithms for piecewise regression of univariate and bivariate data using fixed-degree polynomials. Specifically, given a set S of n data points (x1, y1), . . . , (xn, yn) ∈ Rd × R where d ∈ {1, 2}, the goal is to segment xi’s into some (arbitrary) number of disjoint pieces P1, . . . , Pk, where each piece Pj is associated with a fixed-degree polynomial fj : Rd → R, to minimize the total loss function λk+􏰄ni=1(yi −f(xi))2, where λ ≥ 0 is a regularization term that penalizes model complexity (number of pieces) and f : 􏰇kj=1 Pj → R is the piecewise polynomial function defined as f|Pj = fj. The pieces P1,...,Pk are disjoint intervals of R in the case of univariate data and disjoint axis-aligned rectangles in the case of bivariate data. Our error approximation allows use of any fixed-degree polynomial, not just linear functions. Our main results are the following. For univariate data, we present a (1 + ε)-approximation algorithm with time complexity O(nε log1ε), assuming that data is presented in sorted order of xi’s. For bivariate data, we √ present three results: a sub-exponential exact algorithm with running timemore »nO( n); a polynomial-time constant- approximation algorithm; and a quasi-polynomial time approximation scheme (QPTAS). The bivariate case is believed to be NP-hard in the folklore but we could not find a published record in the literature, so in this paper we also present a hardness proof for completeness.« less