Data Sharing and Compression for Cooperative Networked Control
Authors: Jiangnan Cheng
Marco Pavone
Sachin Katti
Sandeep P. Chinchali
Ao Tang
Venue: NeurIPS 2021 Poster (2021)
Keywords: Decision and Control
Representation Learning
Applications (Societal Systems)

Abstract: Sharing forecasts of network timeseries data, such as cellular or electricity load patterns, can improve independent control applications ranging from traffic scheduling to power generation. Typically, forecasts are designed without knowledge of a downstream controller's task objective, and thus simply optimize for mean prediction error. However, such task-agnostic representations are often too large to stream over a communication network and do not emphasize salient temporal features for cooperative control. This paper presents a solution to learn succinct, highly-compressed forecasts that are co-designed with a modular controller's task objective. Our simulations with real cellular, Internet-of-Things (IoT), and electricity load data show we can improve a model predictive controller's performance by at least 25% while transmitting 80% less data than the competing method. Further, we present theoretical compression results for a networked variant of the classical linear quadratic regulator (LQR) control problem.

Data Sharing and Compression for Cooperative Networked Control
Jiangnan Cheng', Marco Pavone", Sachin Katti', Sandeep Chinchali', and Ao Tang'
School of Electrical and Computer Engineering, Cornell University, Ithaca, NY Department of Aeronautics and Astronautics, Stanford University, Stanford, CA Department of Computer Science. Stanford University. Stanford, CA *Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX edu,
{jc3377,
edu, {pavone, edu, sandespc@utexas.ee edu
Abstract
Sharing forecasts of network timeseries data, such as cellular or electricity load pat- terns, can improve independent control applications ranging from traffic scheduling to power generation. Typically, forecasts are designed without knowledge of a downstream controller's task objective, and thus simply optimize for mean pre- diction error. However, such task-agnostic representations are often too large stream over communication network and do not emphasize salient temporal features for cooperative control. This paper presents a solution to learn succinet, highly-compresee forecasts that are co-designed with a modular controller's 's task objective. Our simulations with real cellular, tntemne--of-Thngge (IoT), and electric- ity load data show we can improve a model predictive controller's 's performance by at least 25% while transmitting 80% less data than the competing method. Further, we present theoretical compression results for a networked variant of the classical linear quadratic regulator (LQR) control problem.
1 Introduction
Cellular network and power grid operators measure rich timeseries data, such as city-wide mobility and electricity demand patterns. Sharing such data with external entities, such taxi fleet operator, can enhance a host of societal-scale control tasks, ranging from taxi routing to battery storage optimization. However, how should timeseries owners represent their data to limit the scope and volume of information shared across a data boundary, such as a congested wireless network:]
At first glance, it might seem sufficient to simply share generic demand forecasts with any down- stream controller. Each controller, however, often has unique cost function and context-specific sensitivity to prediction errors. For example, cell demand forecasts should emphasize accurate peak-hour forecasts for taxi fleet routing. The same underlying cellular data should instead emphasize fine-grained throughput forecasts when video streaming controller starts download. Despite the benefits of customizing forecasts for control, today's 's forecasts are mostly task-agnostic and simply optimize for mean median prediction error. As such, they often waste valuable network bandwidth to transmit temporal features that are unnecessary for a downstream controller. Even worse, they might not minimize errors when they matter most, such as peak-hour variability.
Given the limitations of today's task aank-tgn forecasts, this paper contributes a novel problem formulation for learning task-driven forecasts for networked control. In our general problem (Fig.
"Uher processes petabytes of data per day ① and a mohile operator can process 60 TB of daily cell metrics Even a fraction of such data is hard to send.
2
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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D. an operator measures timeseries St, such as electricity or cell demand, and transmits compressed representation Φi, which is decoded 8, at the controller. Rather than simply minimize the prediction eroor for 8t, we instead learn a representation that minimizes modular controller ' ultimate cost J. Our key technical insight is to compute a controller's sensitivity to prediction errors, which in turn guides how we co-design and learn a concise forecast representation that is tailored to control. As such, our scheme jointly integrates data-driven forecasting, compression, and model-based control.
Related work: Our work is broadly related to compression for control as well as task-driven representation learning. The closest work ours B3, where task-driven forecasts are learned for one-step stochastic op- timization problems. In stark contrast, we ad- dress compression of timeseries forecasts and focus on networked, multi-step control prob- lems. Our work is also inspired by Shannon's rate-distortion theory [4], which describes how to encode and transmit signals with minimal bit-rate to minimize reconstruction error. In con- trast, work with real numbers rather than bits and focus on reducing the dimension of data while keeping task-specific control cost low.
Timeseries Generator
Control System
Encoder
Forecast Decoder
V
St-W
4
Pr
Bg
Controller
Cost
plant State
Control Nr
Data sharing for cooperative control: An owner of timeseries data st, such as mobile operator, needs to transmit a compressed represen- tation o to downstream controller with internal state Ft- The learned forecast emphasizes task- relevant temporal features to minimize end-to-end controller cost J.
Figure 1:
Prior work has addressed rate-distortion trade- offs for networked LQR control problems [5]- 77. However, these works focus on ensuring closed-loop stability for remote controller and a physically-separated plant, such as in tele- operation. Our problem is fundamentally dif- ferent, since we address how external timeseries forecasts can enhance a controller's local decisions using full knowledge of its own internal state. While the term co-design appears in select work on networked LQR, refers to a drastically different setting where a communication scheduler and tele-operated controller must be jointly designed [8-II. Moreover, event event- triggered control/learning 12:15 emphases temporal sparsity of communications, while in our setting the MPC controller consistently requires forecast of timeseries. Finally, work differs from deep neural network
(DNN) compression schemes for video inference [16 17
since we focus on control.
More discussions will follow in Sec. 2 after the technical problem is introduced in detail.
Contributions: In light of prior work, our contributions are three-fold. First, we introduce a novel problem for learning compressed timeseries representations that are tailored to control. Second, to gain insights into our problem, we contribute analytic compression results for LQR control. These insights serve a foundation for our general algorithm that computes the sensitivity of model predictive controller (MPC) to prediction errors, which guides learning of concise forecast representations. Third, we learn representations that improve control performance by 25% and are 80% smaller than those generated by standard autoencoders, even for real loT sensor data we captured on embedded devices as well as benchmark electricity and cell datasets.
Organization: In Sec. [ we formalize general problem of compression for networked control and provide analytical results for LQR. Then, in Sec. 3 we contribute an algorithm for task-driven data compression for general MPC problems. We demonstrate strong empirical performance of our algorithm for cell, energy, and IoT applications in Sec. and conclude in Sec. D
Problem Formulation
We now describe the information exchange between generator of timeseries data, henceforth called
a forecaster, and a controller, as shown in Fig. [ Both systems operate in discrete time, indexed by t,
for time horizon of T steps. The notation Yeub denotes timeseries y from time a to b.
Forecast Encoder: The forecaster measures a high-volume timeseries 8, E RRP Timeseries drawn from domain-specific distribution D, such as cell-demand patterns, denoted by So:T-1 D.

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A differentiable encoder maps the past W measurements, denoted by st-W+1a- to a compressed representation &, R using model parameters 8g o, and is referred to as the bottleneck dimension since it limits the communication data-rate and how many floating-poing values are sent per unit time.
f.). Typically, Z < p
The compressed representation de is transmitted over a bandwidth-constraine communication network, where a downstream decoder maps Φe a forecast $1:+ for the next Il steps, denoted by: Se.e+ - Ieecode (Ør; 0a), where 0a are decoder parameters. Importantly, we decode representation Φe into a forecast & so it can be directly passed to prede--predictve controller that interprets & as physical quantity, such as traffic demand. The encoder and decoder jointly enable compression and forecasting by mapping past observations a forecast via bottleneck Φz.
Forecast Decoder:
Modular Controller: The controller has an internal state 24 E II and must choose an optimal control We E IRT We denote the admissible state and control sets by X' and U respectively. The system dynamics also depend on external timeseries & and are given by: T++1 = f Ue, 8t), t E (O. T- 1}. Importantly, while state 2; depends on exogenous input st, we assume S; evolves independently of 24 and ug. This is a practical assumption in many networked settings. For example, the demand 5, for taxis might mostly depend on city commute patterns and not an operator's routing decisions or fleet state ry. Ideally, control policy chooses a decision based on fully-observed internal state 24 and perfect knowledge of exogenous input -t: Uq T (Iz, #1: 8e).
where d are control policy parameters, such as a feedback matrix for LQR. However, in practice, given a possibly noisy forecast $ 1-11 it will enact a control denoted by u, = π & #1 0z). which implicitly depends on the encoder/decoder parameters 0c,, Pa via the forecast 8.
Our main objective is to minimize end-to-end control cost J which depends on initial state zo and controls -0 which in turn depend on the forecast 80 For a simpler notation, we use bold variables define the full timeseries, i.e., := uo:T-1, : S0:7-1, û = 00:T-1 and 8 == S1 The control cost J is a sum of stage costs c(2,, ûe) and terminal cost (TT)) (ū; zy s) (xT) Ei=o c(2,, ûz), where I++1 = f(z, in st), t {O..... T 1}. Importantly, the above plant dynamics ff evolve according to true timeseries St, but controls ún are enacted with possibly noisy forecasts $4.
Control Cost:
In practice, a designer often wants to visualize decoded forecasts debug anomalies or view trends. While our principal goal is to minimize the control errors and cost associated with forecast 8, we allow a designer to optionally penalize mean squared prediction error (MSE). This penalty incentivizes a forecast 8, to estimate the key trends of st. serving as regularization term: JF (s. + E ||s, 11...
Forecasting Errors:
Overall Weighted Cost: Given our principal objective of minimizing control cost and optionally penalizing prediction error, we combine the two costs using user-specified weight XF Importantly, we try to minimize the additional control cost J((ū; zi, s) incurred by using forecast $ instead of true timeseries s, yielding overall cost:
joot. (u. ú. 20, T (J 20, s) ( J zo; s)) + X JF (s
(1)
extra control cost
The total cost implicitly depends on controller, encoder, and decoder parameters via controls u and û and the forecast S. Having defined the encoder/decoder and controller, we now formally define the problem addressed in this paper.
Problem 1 (Data Compression for Cooperative Networked Control). We are given controller x1; 0o) with fixed, pre- pre-trained parameters ee. fixed bottleneck dimension Z, and perfect measurements of internal controller state Tu:r- Given true exogenous timeseries So:1-1 drawn from data distribution D. find encoder and decoder parameters 0e, Ga to minimize the weighted control and forecasting cost (Eq. 7h with weight X²:
8% 03 = argmin EnoT-r-D(Jooh (u, (,, S, A5)J, where 8a
Φ = Yencode (8:- W414; 0c), di ER²
$ + f1 1 = Idvcode Φ6 Ba),
44 = n (It, Stst 11 1;)
T 8t4 (ttt
It+1 fzr, 8e, St), and 21 X, EU, t (0...., .T -T-13 1}.
3

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Technical Novelty and Practicality of our Co-design Problem:
Having formalized our problem,
now articulate how it differs from classical networked control
and tele-operation [18 [6] 5), compressed sensing [11 22). and certainty-equivalent control First, we can not readily apply the classical separation principle [25] of Linear Quadratic Gaussian (LQG) control, which proscribes how to independently design a timeseries estimator, such controller (the linear quadratic regulator) for optimal performance. This is because the timeseries owner measures a non-stationary timeseries &t (e.g. spatiotemporal cell demand patterns), without an analytical process model for standard Kalman Filtering, motivating our subsequent use of learned DNN forecasters. Second, due to data-rate constraints, we must prioritize task-relevant features as opposed equally weighting and sending the full s, which a classic state observer in LQG would do.
23 24
as the Kalman Filter [26], and "certainty-equivalee
Moreover, when the estimator and controller separated by bandwidth-limited network and the separation principle does not hold [27]. our setting still differs from classical networked control [8-20 6 5]. These works assume that both the full plant state x; and controls Ue are encoded and transmitted between a remote controller and plant. In stark contrast, the only transmitted data in our setting is external information 9, from network operator, which can improve an independent controller's local decisions based on its internal state zt. As such, simply grouping controller state Iy and network timeseries $; into a joint state for classical tele-operation is infeasible, since 2; and s are measured at different locations by different entities. In essence, Prob. [formalizes how network operator can provide significant value to independent controller by judicious data sharing.
Forecaster and Controller Co-design
Prob. iis of wide scope, and can encompass both neural network forecasters and controllers. For intuition, we first provide analytical results for an impur-driver LQR problem in Sec. 3.I We then such insights in general learning algorithm that scales to DNN forecasters in Sec. 3.2
3.1 Input-Driven Linear Quadratic Regulator (LQR)
We first consider a simple instantiation of Prob. ! with linear dynamic no state or control constraints, and quadratic control cost. Since the dynamics have linear dependence on the exogenous input we refer to this setting as an input-driven LQR problem. We first analyze the problem when controls computed for the full-horizon from 0 to T H and then extend receding- control (MPC) in Sec. 3.2 The dynamics and control cost are:
Jt+1 Ax, + Bui + Cats
H
H-
J
Σ 2 Qu. 1=
+
Σ
Ruis
(3)
t=0
where Q, R are positive definite. Our first step is to determine the optimal control. Given the lincar dynamics, for all times (0,.. ... H 1). each 1441 is linear function of initial condition vn and the full future control vector u and s:
Fit1 = A do + M,U + Nas, where
(4)
M, [A'B A'
B
D R N = [A'C A'-CC
C
0
BaxpH
Therefore, given Tn and vector s, control cost J is quadratic function of u:
#-1
h
(u Fo. s) = u +
Σ
M, QM,) 2
Σ M, Q(A4+1+2 20 NNs) T u+ constant,
k(xo.s)
(5)
Transition noise is not added here due to certainty equivalence of input-driven LQR, as shown in Appendix
Sec. A.2

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where the constant of Ei=0 (A*+) To + N₁s) T Q(A*** Hо + N₃s) is independent of u, and R blockdiag( R, ...... R) E
R Rn xmll Clearly, K is positive definite and J is strictly convex. Given the convex quadratic cost, the optimal control is u", where u K K- s). However, given possibly noisy forecast 6, we would instead plan and enact controls denoted by ú, where ú --- (Kx,, Thus, the sensitivity of such controls forecast errors is:
-
H-1
-K (k(0, 8) k( s)) = K-
Mj QN::) s),
(6)
and the sensitivity of the control cost to forecast errors is:
(ú; no, s) no. s) = (ū u²) K(ū - n²) =
L LKK-L s):
(7)
co-design matrix V
where we term the positive semi-definite co-design matrix V L K L. We now combine the extra control cost and prediction error to calculate the total cost as:
1 H ((S (B-s)) S (V+ s)) (8)
Jio..
s) ( s)
+
S
S)
control
prediction error
The above expression leads an intuitive understanding of co-design. The co-design matrix v in Eq. essentially weights the error in elements of š based on their importance to the ultimate control cost. Thus, our approach is fundamentally lask-aware since the co-design matrix depends on LQR's dynamics, control, and cost matrices shown in Eq. [6] and 7 The optional weighting of prediction error with F acts regularization term. Moreover, we now show that we can reduce input-driven LQR to a low-rank approximation problem, which allows to find an analytic expression for an optimal encoder/decoder.
Imput-Drived LQR is Low-Rank Approximation. Given the above expressions for the total cost, we now assume simple parametric model for the encoder and decoder to formally write Prob. [ for the toy input-driven LQR setting. Specifically, we assume a linear encoder E R Zxxx/ maps true exogenous input s to representation
Es, where E IR?. Then, linear decoder matrix D E R yields decoded timeseries & DΦ = DEs. In practice, we often have training dataset consisting of N samples of exogenous input s drawn from a data distribution s D. These samples can be arranged as columns in matrix S E TR To learn an encoder E and decoder D from N samples once, we can express our problem as:
argmin D.E
Σ -S) ( * S.). where
DES, rank(D) < Z and rank(E) S,
(9)
where S; and S represent the i-th column vector of S and S. We now characterize the input-driven LQR problem.
Proposition 1 (Linear Weighted Compression), Input-drived LQR (Eq. 9 a low-rank approxima- tion problem, which admits an analytical solution for optimal encoder and decoder pair (E, D).
Proof. We first re-write the objective of the input-driven LQR problem (Fq. 9 as: (S,
S,) T (Y S,) J A where YAY is the eigen ligen-decomposition of the positive definite matrix V + F F and ||-I|r represents the Frobenius norm of matrix. Thus, the problem can be written as:
argmin D.E
A DES
ALY
11 where rank(D) < Z and rank( (E) S Z,
(10)
appeoximation
original
which is the canonical form low-rank approximation problem. By the Eckhart- Younn theorem, the solution to the input-driven LQR problem (Eq. [00 is the rank Z truncated singular value decomposition (SVD) of original matrix T S. denoted by U.VT. In the truncated SVD,
5

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U E Rolii? 2 is semi semi-orthogonal, Σ E R is diagonal matrix of singular values, and V E R is semi- cemi-orthogonal. Further, an encoder of F = = and decoder of D (A solve the problem since:
DES
DES = T (ARY UU A ((UTARYTS) -
VEV
approximation
E
optimal rank 2 approximation
A similar analysis for linear encoder-decoder structure for networked inference, not control, is presented in [7]. The key difference from our current paper is our problem setup is for control, not networked inference. Moreover, our result differs from existing LQR literatures with exogenous input, such as (28), since our exogenous input 5, is subject to to a network bottleneck and encoder/decoder, which is the crux of our Prob. [ and our total cost includes the extra control cost due to mis-estimation of s,, rather than simply the prediction error 8t.
Compression benefits: Casting input-driven LQR as low-rank approximation provides significant intuition. As shown in Proposition [ the optimal encoder/decoder depend on the truncated SVD of A S, which takes into account the control task via the co-design matrix, importance of prediction errors via X", and statistics of the input via S. We achieved strong compression benefits for simulations of input-driven LQR (provided supplement Fig. 5 due to space limits).
Transitioning to Model Predictive Control (MPC). In practice, we often have forecasts for short horizon H T. Then, starting from any state 2;, MPC will plan sequence of controls 1t:- H-11 enact the first control it and then re-pla re-plan with the next forecast. If we replace the horizon to H < T in the input-driven LQR analysis in Sec. 3.1 u short-horizon H and we can encode/decode using low rank approximation as in Prop. 1. While performance is not necessarily optimal for the full duration MPC performs extremely well in practice, yielding even better compression gains, as shown in the supplement (Fig. (
K 'k s) gives the optimal control for
We also note practitioner can adopt a sim- ple cost function based on MPC that comple- ments Eq. D The MPC controller optimize the cost foot. given a short-horizon forecast $ but only enact the first con- trol ú, = π (Ix, St:t+ H H-1; 02). Mcanwhile, the best first control MPC can take u, = T (It, St -1: 0e) with perfect knowledge of for horizon H. Thus, our insight that we can penalize the errors in enacted controls ún during training and regularize for prediction error, us- ing cost: + Et= || 1 + sy sx In experiments, we observed strong perfor- mance by optimizing for the cost Eq. [ as well as the above cost, which optimizes fiot. over a short-horizon for MPC. We now crystallize these insights from input-driven LQR into for- mal algorithm that applies to data-dri data-driven MPC.
will
Algorithm 1 Compression Co-design for Control
1: Set forecast weight XF, bottleneck size Z
2: Initialize encoder/decoder parameters 00 00 randomly, and fix controller parameters 0.
3: for 5-0 to Nepuch
do
4: Initialize Controller State zo E X
5: fort 0 to T 1 do
6: Encode Φ; = Genendo (s- w 0%)
7:
Decode 81:1 = Adocode (#r; (g) Enact 0, T T(I,, 111 00)
8:
9:
Propagate J2+1 + [ Ug 81)
10:
Un = T(I,, S10tH-1; (For Training Only)
11:
end for
gT+1 0
BACKPROP joo (u. ú s, 20 2......))) XF)]
13: end for
3.2 Algorithm
Return learned parameters Nap 0 Nawa
to Co-design Forecaster and Controller
For more complex scenarios than LQR, it challenging provide analytical forms of an optimal encoder and decoder. Thus, we present heuristic algorithm to solve Prob. in Algorithm Our key technical insight is that, if the encoder, decoder, and controller are differentiable, we can write:
V tot. (u, û. S, S; æo. 2²)
V u, û. S, 20 AB)
( -)
X
(II)
40.-
s S)
v0.

---

and likewise for 8a. The first term captures the sensitivity of the control cost with respect to prediction errors and the second propagates that sensitivity to the forecasting model. Crucially, the gradient of Joot. can be obtained from recent methods that learn differentiable MPC controllers 29 30
In lines 1-2 of A we randomly initialize the encoder and decoder parameters and set the latent representation size Z to limit the communication data-rate. Then, we enact control policy rollouts in lines 3-11 for Nepoch training epochs, each of duration T.. We first encode and decode the forecast 8 (lines 6-7) and pass them to the downstream controller with fixed parameters Og (lines 8-10). During training, we calculate the loss by comparing the optimal weighted cost with true input and the forecast $. In turn, this loss is used to train the differentiable encoder and decoder through backpropagation in line 12. Finally, the learned encoder and decoder (line 14) are deployed.
Co-design Algorithm Discussion: A few comments are in order. First, true input s is only needed during training, which is accomplished at single server using historical data to avoid passing large gradients over a real network. Then, we can periodically re-train the encoder/decoder during online deployment. Second, our approach also applies when Og are parameters of a deep reinforcement learning (RL) policy. However, since the networked systems we consider have well-defined dynamical models, we focus our evaluation on model-based control.
4 Application Scenarios
We now describe three diverse application scenarios addressed in our evaluation. The scenarios are linear MPC problems with box control constraints:
It+1 = Zz + U - St, (Dynamics) where
Umin S Ue S Umax (Constraints)
(12)
Our scenarios have the same state and control dimensions m = n,, and dynamics/control matrices A B- Inxn indicate uniform coupling between controls and the next state. Finally, have actuation limits Umin and Mmax- The cost function incentivizes regulation of the state x, to a set-point L. In practice, we often want to penalize states below the set-point, such inventory shortages where 2t < L. more heavily than those above, such as excesses. In the following cost, weights Ye: Yes Tu E 1= govern excesses, shortages, and controls un respectively:
T J(((, E (1elll - L]-112 + 7 111+ +
Σ 1=
(13)
1=0
where [x]+ represents the positive elements of a vector. We focus on linear MPC with box constraints
and flexible quadratic cost (Eq, [3) since it is canonical problem [31] 32] with wide applications
in networked systems. However, to show the generality of co-design, we provide strong experimental results for mobile video streaming application with noisy, non-linear dynamics in Sec. 5.2 We evaluate diverse MPC settings coupled with an array of neural network forecasters.
We consider an idealized scenario similar to datacen- ter temperature control [33], where x; E R represents the temperature, humidity, pressure and light for & machines in a smart factory, each of whose 4 sensor measurements we want to regulate to a set- point of L. External heat, humidity, and pressure disturbances s E RD add to state 2; the dynamics (Eq. [2). Disturbances are measured by n IoT sensors, such as from nearby heating units. Our objective is select control inputs u E RM regulate the environment anticipating disturbances from the p loT sensors. The cost function (Eq. 13) has Ve % 7h to equally penalize deviation from the set-point and regulation effort. Finally, we collected two weeks of stochastic timeseries of temperature, pressure, humidity, and light from the Google Edge Tensor Processing Unit (TPU)'s cnvironmental sensor board for our experiments, as detailed in the supplement.
Smart Factory Regulation with 1oT Semsors:
Taxi Dispatch Based on Cell Demand Data: In this scenario, state E IR" represents the difference between the number of free taxis and waiting passengers n city sites, so 2, > 0 represents idling taxis while < O represents queued passengers. Control E IRm represents how many taxis are dispatched to serve queued passengers. Exogenous input s; E R represents how many new passengers join the queue at time t. Of course, the taxi service has historical forecast of 8,, but the cellular operator can use city-wide mobility data to improve the forecast. Our goal is to regulate x; to I, = 0 to neither have waiting passengers nor idling taxis. In the cost function (Eq. [3]. we have 7. = 7, = 100 and 7a = 1 to heavily penalize customer waiting time for long queues. Our simulations use 4 weeks of stochastic cell demand data from Melbourne, Australia from (34).
7

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Battery Storage Optimization: Our final scenario is inspired by closely-related work to ours B] who consider how a single battery must be charged or discharged based on electricity price forecasts. Since our setting involves a vector timeseries 8, we consider electrical load forecasts from multiple markets. Thus, we used electricity demand data from the same PJM operator as in R], but from multiple markets in the eastern USA [35. Specifically, state I2 R represents the charge on n batteries and control Un E 1 represents how much to charge the battery to meet demand. Timeseries 8 E Rº represents the demand forecast at the locations of the n batteries, where p In the cost function (Eq. (3). we desire a battery of total capacity 2L to reach a set-point where is half-full, which, per B3, allows flexibly switching between favorable markets. Further, we set Ye % 7a 1.
5
Evaluation
The goal of our evaluation is to demonstrate that our co-design algorithm achieves near-optimal control cost, but for much smaller representations Z compared to task-agnostic methods.
Metrics. We evaluate the following metries: 1) We quantify the control cost for various bottleneck sizes Z, relative to the optimal cost when ground-trute input 8 is shared without a network bottleneck. 2) To quantify the benefits of sending representation of size Z compared to the full forecast st.++--1 of size pł,, we define the compression gain as Z We also compare the minimum bottleneck Z required to achieve within 5% of the optimal cost for all benchmarks. 3) Since the objective of Prob. [ also incorporates prediction error, we quantify the MSE forecasting error for various Z.
pH
Algorithms and Benchmarks. We test the above metrics on the following algorithms, which represent various instantiations of Alg. for different X as well as today's prevailing method of optimizing for prediction MSE. Our algorithms and benchmarks are: 1) Fully Task-aware (A = 0): We co-design with X 0 according to Alg. | to assess the full gains of compression. 2) Weighted: We instantiate Alg. [ with X > 0 to assess the benefits of task-aware compression as well as forecasting errors induced by compression. In practice, X is user-specified. For visual clarity, we show results for X = 1 in Fig.
2 since the trends for other X mirror those in Fig. 5 3) Task-agnostic (MSE): Our benchmark learns forecast & to minimize MSE prediction error, which is directly passed to the controller without any co-design.
Forecaster and Controller Models. We compared forecast encoder/decoders with long short term memory (LSTM) DNNs (36] and simple feedforward networks. We observed similar performance for all models, which we hypothesize is because co-design needs to represent only small set of control-relevant features. We used standard DNN architectures, hyperparameters, and the Adam optimizer, further detailed in the supplement. Our code and data publicly available at https //github.com/ /oong//ccnjiiie wwngaa//coooprratveeeeooo wttp:/vwwweeetereeeeeooeo control
5.1 Linear Dynamics
We now evaluate our algorithms the IoT, taxi scheduling, and battery charging scenarios described in Sec. 4 Our results on a test dataset are depicted in Fig. 2 where each column corresponds to a real dataset and each row corresponds to an evaluation metric, discussed below.
How does compression affect control cost? The first row of Fig. quaatifies the control cost J for various compressed representations 2. The optimal cost, in dashed black line, is an unrealizable lower-bound cost when the controller is given the true future 8t.t+ 1-1 without any forecast error. The vertical bars show the distribution of costs across several test rollouts, each with different timeseries S. Our key result is that our task-aw aaskaware scheme (orange) achieves within 5% of the optimal cost, but with small bottleneck size Z of 4, 4 and 2 for the 1oT, traffic, and battery datasets, respectively. This corresponds to an absolute compression gain of 15x, , 15x, and 96x for each dataset. In contrast, with the same bottleneck sizes, competing task-agnostic scheme (blue) incurs at least 25% more control cost than our method.
Moreover, for the IoT and battery datasets, the task agnostic benchmark requires large bottleneck of Z = 35 and Z = 11, leading our approach to transmit 88% and 82% less data respectively. Strikingly, even for large representation of 2 60, a task-agnostic scheme incurs 100% more cost than the optimal for the cell traffic dataset. This is because the cost function is highly sensitive to shortages
with 7a Ne. which is not captured by simply optimizing for mean error. To clearly see the trend
8

---

Optimal
Task-agnostic
Task-aware, =
Weighted, ===1.0
Control Cost
Control Jost
353
Control Cost
303
800
150
899
100
2
Bottleneck Dimension
Bottleneck Dimension
Bottleneck Dimension Z
Control Errer Errr (MSE)
0.20
Control Error (MSE)
0,020
Control Error (MSE)
8.015
0.02
0.59
8.055
8.81
0.50
0.08
Timeseries Element u(i)
Timeseries Element u(i)
Timoseries Element u(i)
Forecasting Emor (MSE)
Forecaeting Error Jf (MSE)
Error (MSE)
0.00
0.90
0.86
0.014
0.05
2.04
8.04
0.800
2.03
0.94
0.000
0.02
0.006
0.22
0.01
Relative Time Index after
Relative Time Index after
Relative Time Index after
Figure 2: Real-world dataset results: From left right, the columns correspond smart factory regulation from IoT sensors, taxi dispatching with cell demand, and battery storage optimization. (Row 1) Co-design achieves lower cost J smaller bottlenecks compared to task-agnostic methods. (Row 2) We also achieve lower error for each dimension i of the vector control, u(i), plotted for a highey-compressed = 3. (Row 3) Co-design heavily reduces forecasting errors for initial horizons that are especially important for MPC's decision decision-making.
in Fig. 2 only plot until 7 9, but the experiments until Z 60. Our weighted approach (green) requires a marginally larger representation than the purely task-aware approach (A = 0) since should minimize both control and forecast error.
Does co-design reduce control errors? We now investigate how the compression benefits co- design arise. Given the stochastic nature of all our real world datasets, all prediction models inevitably produce forecasting error, which in turn induce errors selecting controls. However, the key benefit of co-design methods is they explicitly model and account for how MPC chooses controls based on noisy forecasts 8, and are thus able to minimize the control error, which we now quantify.
As defined in Sec. 3.1 for any state Ue is the optimal first MPC control given perfect knowledge of st:t--11+ while ūe is MPC's actual enacted control given a noisy forecast. Then, the control errors across various control dimensions i are the MSE error ||u, (i) û (i) between optimal control (i) and ū,(i), The second row of Fig. 2 clearly shows that our task-aware and weighted methods (orange and green) achieve lower control eπror all three datasets.
Why does co-design yield task-relevant forecasts? reduces forecasting error for the purposes of an ultimate control task, we show forecasting errors across various time horizons in the third row of Fig. 2 As argued in the previous section, all forecasting models produce prediction error. However, task-agnostic forecast (blue) roughly equally distributes prediction error across the time horizon to + H In stark contrast, weighted co-design approach (green) drastically reduces prediction errors in the near future since MPC enacts
further show that our co-design approach

---

the first control and then re-plans on rolling horizon. Of course. the full forecast $t.++--11 matters enact control plan #1 but the cost is most sensitive the initial forecast and control errors in our MPC scenarios. For visual clarity, we present forecast errors of the fully task-aware approach (A = 0) in the supplement, since the errors are much larger than the other two methods.
5.2 Nonlinear Dynamics with Transition Noise
To illustrate that our co-design approach works well for systems with nonlinear and stochastic dynamics, we also provide a nonlinear example concerning idealized mobile video streaming scenario. In this application, a mobile video client stores buffer of video segments and must choose a video quality to download for the next segment of video. The goal is maximize the quality of video while minimizing video stalls, which occur when the buffer under-flows while waiting for segment to be downloaded. Here, state 2, represents the buffer of stored video segments, control is segment quality, and st network throughput. The nonlinear dynamics are 2++1 = [z)
Optimal
Task-agnostic
Tark-aware, 2"=0 Weighted,
Control Cost J
1.8
1.7
Bottleneck Dimension
@ st]+ + Izz + ?ht where 2 represents element-wise division, La is the increase in stored video for each download, and Th is Gaussian transition noise. The cost aims to keep a positive buffer and have high video
Figure 3: Co-Design Results with Nonlinear Dy- namics and Transition Noise.
quality: [x(,, u) Et=o T/ll Ten EED
7h | T. 113
Fig. elearly shows our approach works quite well for nonlinear scenario with transition noise,
which complements the three linear examples in Sec. [. In the above experiments, the parameters are: T = 60,
H 15, m n = = 4, Th 0.25, The 1, Lg = 0.5 x 1,, Ly = 0.2 1
Limitations: Our work does not automatically learn the optimal bottleneck size Z that minimizes control cost nor necessarily leam human iamaaninterpretable latent representation.
Conclusion
Society is rapidly moving towards "smart cities" (7.38) where smart grid and 5G wireless network operators alike can share forecasts enhance external control applications. This paper presents preliminary first step towards this goal, by contributing an algorithm to learn task-relevant, compressed representations of timeseries for control objective. Our future work will center around privacy guarantees that constrain learned representations to filter personal features, such as individual mobility patterns. Further, internal states of the downstream controller. While recent work has addressed how to value datasets for supervised learning 33 40 a promising extension of work is to price timeseries datasets for cooperative control in data-market. Indeed, our ability to gracefully trade-off control cost with data exchange lends itself an economic analysis.
certify algorithm does reveal proprietary control logic private
7 Funding Disclosure
This material is based upon work supported by the National Science Foundation under Grant No. 2133481 and 2133403. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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12


Reviews:
[{"review_id": "qOz5f1EoDh", "reviewer": "Reviewer_zhDr", "summary": "This paper studies sending compressed representations for cooperative control by creating task-driven representations that are co-designed with the control task. They present theoretical compression results (of LQR being low-rank compression in prop 1) and go on to study non-linear cellular, IoT, and electricity load data settings. The idea, summarized in Algorithm 1, is to parameterize encoders/decoders, use the decoded signal for optimization, and optimize the loss in Eq 8 that includes the control loss and a prediction component.", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "This is a well-motivated and timely study of end-to-end learning for cooperative control. They take the existing pieces of differentiable control and end-to-end model learning and theoretically and empirically study them in a new cooperative context. To the best of my knowledge, the theoretical results in the LQR setting are novel, and the experiments in Fig 2 validate the idea that including the downstream signal enables more efficient latent representations of the data to be learned. The battery storage optimization task from [3] is reasonable extended from the single to multiple battery setting. My largest experimental concerns is that the bottleneck dimensions used in Fig 2 are relatively small and only consider dimensions <10. Realistically, the bandwidth in these settings should allow for higher-dimensional data to be transmitted.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The paper does not significantly discuss these and it could be insightful to discuss scalability limitations on how scaling the bottleneck dimension and control problem sizes impact the overall tractability/learnability.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "1 hour"}, "extra_text": {}}, {"review_id": "ga-L_BBJqPh", "reviewer": "Reviewer_vpGi", "summary": "This paper presents a solution to learn succinct, compressed time-series embeddings, which also considers downstream modular controller’s task objective.\n\nContributions:\n1. novel problem: learning compressed time series representations which are also tailored to the downstream control tasks.\n2. Analytic compression results on LQR control for illustrations on the problem.\n3. Experiments on real-world datasets", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Originality: This paper studies a novel task that learns time series representations for the downstream control tasks. This paper discussed clearly how this work differs from previous contributions on page 3,  with the most relevant paper cited.\n\nQuality: The submission is technically sound and the claims are well supported by theoretical analysis on the  LQR problem. Then the paper expands to more general settings like DNN forecasters. \n\nClarity: The submission is clearly written and well organized.\n\nSignificance: The results are important, not only from the perspective of improvements on baselines but also from the perspective of possible application domains. This paper discussed the diverse application scenarios like smart factory on IoT sensors, taxi dispatch, and battery storage. ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors pointed out that they do not automatically learn the optimal bottleneck size that minimizes control cost nor necessarily learns a human-interpretable latent representation.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2"}, "extra_text": {}}, {"review_id": "Bz3dkpRCCAb", "reviewer": "Reviewer_rL7L", "summary": "This paper proposes an architecture and gradient-based training for an encoder/decoder scheme to reduce communication of an exogenous signal to a remote optimization-based controller. Such scenarios are given, for example, when a remote location has forecast information (e.g., traffic) that is relevant for an optimization-based controller (e.g., optimal scheduling of taxis). The goal of the paper is to reduce communication of the exogenous, while maintaining reasonable performance. The main idea is to take the control objective into account when training the compression scheme, so that the encoder/decoder is optimized for the task at hand. The claimed contributions are (i) introduction of a novel problem, (ii) analytical results for the LQR case, and (iii) improved performance and compression on real-world data sets.", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "## Originality\n\n1) The components of the presented approach (encoder/decoder for reduced communication, gradient-based end-to-end training, remote control architecture) are not new. Also the idea to take a downstream task into account when designing reduced communication schemes is not entirely new (see next comment). Nonetheless, this reviewer is not aware of the proposed combination of techniques. Thus, I see the main contribution in that combination and the demonstration that this leads to reduced communication.\n\n2) Related work: The discussion of related work is mostly adequate.  However, I was missing a *specific* discussion on the technical results for the LQR case with exogeneous input in Sec. 3.1.   The LQR problem with exogenous inputs has been studied before - so, how are the technical results derived in Sec. 3.1 related to results known in control?  Just to be clear - the combination of results is likely new, but still, it would be good to see a discussion on how the obtained equations, e.g. for the optimal controller, compare to existing works. \n\nI have further suggestions for the discussion of related work:\n\n2.1) The idea to reduce amount of network traffic by transmitting only the data that is relevant for a certain task has also been investigated under the terms of \"event-triggered communication/estimation/control\" (see surveys Heemels et al, An introduction to event-triggered and self-triggered control; or Miskowicz, Event-based control and signal processing) and, more recently, \"event-triggered learning\".  For example, Schlueter et al. (Event-triggered Learning for Linear Quadratic Control) also use LQR control cost to trigger learning of a better model when needed. These ideas have also been applied to reduce network communication in wireless sensor systems with savings similar to those reported herein (e.g., Beuchert et al, Overcoming bandwidth limitations in wireless sensor networks by exploitation of cyclic signal patterns: An event-triggered learning approach) . While the mentioned event-triggered approaches emphasize sparsity in time, the manuscript at hand has a different perspective on reducing communication.  Thus, the works are not conflicting (maybe orthogonal?), yet are closely related on the level of motivation and application to reducing communication in control networks. In fact, a combination of the approaches might be interesting for further research.  \n2.2) When reading the \"Related Work\" subsection at the beginning of page 2, I was surprised to see no discussion on teleoperation, e.g. in robotics.  The authors later add such a discussion (to some extent), so it might be worth pointing out already at the end of the \"related work\" subsection, that more discussion will follow after introducing the technical problem in detail.\n\n\n\n## Quality\n\nThe paper seems technically sound.  Limitations are briefly discussed at the end of Sec. 5., and in the conclusion.  The numerical examples serve to illustrate main claims of the paper.  IMHO, the experimental results could, however, be improved by reporting on nonlinear examples:\n\n3) The numerical examples presented in the main paper all include linear dynamics.  Obviously, these are less challenging then nonlinear problems.   For a nonlinear problem, the reader is deferred to the supplementary material.  I would find it much more convincing if the nonlinear example was part of the main paper. \n\n\n\n## Clarity\n\nI found the submission clear.  I only have a minor comment on the abstract:\n\n4) In the abstract, \"forecasts are designed without knowledge of a downstream controller’s task objective, and thus simply optimize for *mean* prediction error.\"  -- I don't see this line of argumentation.  Why does the ignorance of a downstream task give rise to optimizing the MEAN only?  In fact, as one does not know what downstream task will be performed, one might want to optimize for predicting full distributions (even though this is hard, of course).\n\n\n\n## Significance\n\n5) As stated above, I see the main contribution in the combination of existing techniques.  To my knowledge, the concrete setting and architecture is novel.  Though, I did not find it too surprising that one can train an encoder/decoder in such a way to save communication. The empirical results demonstrate improvement over more standard ways to train encoder/decoder (without knowledge of the downstream task), yet they don't compare to alternative ways to save communication (e.g., see works mentioned under 2)), so it is hard to judge how significant the communication savings are.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes, IMO.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "~3h"}, "extra_text": {}}, {"review_id": "B4hT_bR5vG3", "reviewer": "Reviewer_ZtkH", "summary": "The paper discusses an approach for deterministic optimal control under estimates of covariate states.", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Overall, I found the paper a good read and easy to follow. However, I am not sure if there is a significant merit to the paper, as the theoretical work seems to me well known and pretty straightforward. I would have either hoped for a bigger focus in the experimental section, which I would have expected to include a real world control scenario or for more work in a theoretical direction for example by including stochastic dynamics. \n\nOriginality: The task overall is not known to me and I am not sure if there are a lot of real world applications for the scenario. The methods presented in the work are pretty standard and there are not many new theoretical insights presented. To me it was not completely clear how the work positions itself in the field w.r.t. related work.\n\nQuality: I did not find any technical mistakes and the mathematics are correct as far as I could check. \n\nClarity: The submission is nicely written, overall easy to follow and well organized.\n\nSignificance: I am not completely sure if the results are very significant. \n\nAspects I found which were missing from a theoretical standpoint are:\n- The work only discusses a deterministic system and therefore no noisy dynamics. Additionally, a probabilistic model description especially for the estimation of the forecast state is probably very favorable in many scenarios.\n-  For the forecast only point estimates are considered. I think a Bayesian estimate would be more appropriate. As this work does not include any uncertainty about the estimates. However, a controller should probably balance its control decisions if estimates are uncertain.\n- Something I did not completely understand was why the controller for $\\hat{u}$ should have the same parametric form as the one for $u$. What is the motivation for this design, similar to certainty equivalence?\n\nThings I missed from the experimental section:\n- Why is the work only evaluated for the very simple dynamical system? For a synthetic evaluation I would have expected also non-linear system dynamics. \n- I would have appreciated if the system would have been evaluated for a real world control problem.\n- For the  closed form solutions in the linear case, I would have appreciated an evaluation.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "N/A", "review_text": "", "extra_scores": {"ethical_concerns": "N/A", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "6"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes an architecture and gradient-based training for an encoder/decoder scheme to reduce communication of an exogenous signal to a remote optimization-based controller. The proposed method takes the control objective into account when training the compression scheme, so that the encoder/decoder is optimized for the task at hand. 

This paper introduces a (to some extend) novel problem, presents theoretical results for the LQR control special case and  demonstrates improved performance on a set of benchmark tasks. However, the reviewers also find that the proposed method is a mix and patch of different techniques (though the proposed combination seems novel) and the numerical demonstrations could be strengthened by including additional baselines.

This paper tackles an problem that might become increasingly important in the future. The reviewers therefore advocate the acceptance of the work, but assume that the criticized details will be addressed in the revision, including
- a discussion of the stochastic problem variation
- clarification of technical contributions and differences to related work (citing (Singh, Pal, 2017) is not the crucial point here, rather that the discussion is _specifc_  (as drafted in the author's response))
Decision: Accept (Poster)