Data Sharing and Compression for Cooperative Networked Control

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 as a 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 ? 1 At a first glance , it might seem sufficient to simply share generic demand forecasts with any downstream controller . Each controller , however , often has a 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 a video streaming controller starts a download . Despite the benefits of customizing forecasts for control , today ’ s forecasts are mostly task-agnostic and simply optimize for mean or 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-agnostic forecasts , this paper contributes a novel problem formulation for learning task-driven forecasts for networked control . In our general problem ( Fig . 1Uber processes petabytes of data per day [ 1 ] and a mobile operator can process 60 TB of daily cell metrics [ 2 ] . Even a fraction of such data is hard to send . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . 1 ) , an operator measures timeseries st , such as electricity or cell demand , and transmits compressed representation φt , which is decoded to ŝt at the controller . Rather than simply minimize the prediction error for ŝt , we instead learn a representation that minimizes a modular controller π ’ s 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 information-theoretic compression for control as well as task-driven representation learning . The closest work to ours is [ 3 ] , where task-driven forecasts are learned for one-step stochastic optimization problems . In stark contrast , we address compression of timeseries forecasts and focus on networked , multi-step control problems . Our work is also inspired by Shannon ’ s rate-distortion theory [ 4 ] , which describes how to encode and transmit signals with a minimal bit-rate to minimize reconstruction error . In contrast , we work with real numbers rather than bits and focus on reducing the dimension of data while keeping task-specific control cost low . Prior work has addressed rate-distortion tradeoffs for networked LQR control problems [ 5– 7 ] . However , these works focus on ensuring closed-loop stability for a remote controller and a physically-separated plant , such as in teleoperation . 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 , it refers to a drastically different setting where a communication scheduler and tele-operated controller must be jointly designed [ 8–11 ] . Moreover , event-triggered control/learning [ 12–15 ] emphases temporal sparsity of communications , while in our setting the MPC controller consistently requires a forecast of timeseries . Finally , our 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 as a foundation for our general algorithm that computes the sensitivity of a 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 IoT sensor data we captured on embedded devices as well as benchmark electricity and cell datasets . Organization : In Sec . 2 , we formalize a 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 . 4 - 5 and conclude in Sec . 6 . 2 Problem Formulation . We now describe the information exchange between a generator of timeseries data , henceforth called a forecaster , and a controller , as shown in Fig . 1 . Both systems operate in discrete time , indexed by t , for a time horizon of T steps . The notation ya : b denotes a timeseries y from time a to b . Forecast Encoder : The forecaster measures a high-volume timeseries st ∈ Rp . Timeseries s is drawn from a domain-specific distribution D , such as cell-demand patterns , denoted by s0 : T−1 ∼ D. A differentiable encoder maps the past W measurements , denoted by st−W+1 : t , to a compressed representation φt ∈ RZ , using model parameters θe : φt = gencode ( st−W+1 : t ; θe ) . Typically , Z p and is referred to as the bottleneck dimension since it limits the communication data-rate and how many floating-point values are sent per unit time . Forecast Decoder : The compressed representation φt is transmitted over a bandwidth-constrained communication network , where a downstream decoder maps φt to a forecast ŝt : t+H−1 for the next H steps , denoted by : ŝt : t+H−1 = gdecode ( φt ; θd ) , where θd are decoder parameters . Importantly , we decode representation φt into a forecast ŝ so it can be directly passed to a model-predictive controller that interprets ŝ as a physical quantity , such as traffic demand . The encoder and decoder jointly enable compression and forecasting by mapping past observations to a forecast via bottleneck φt . Modular Controller : The controller has an internal state xt ∈ Rn and must choose an optimal control ut ∈ Rm . We denote the admissible state and control sets by X and U respectively . The system dynamics also depend on external timeseries st and are given by : xt+1 = f ( xt , ut , st ) , t ∈ { 0 , · · · , T − 1 } . Importantly , while state xt depends on exogenous input st , we assume st evolves independently of xt and ut . This is a practical assumption in many networked settings . For example , the demand st for taxis might mostly depend on city commute patterns and not an operator ’ s routing decisions ut or fleet state xt . Ideally , control policy π chooses a decision ut based on fully-observed internal state xt and perfect knowledge of exogenous input st : t+H−1 : ut = π ( xt , st : t+H−1 ; θc ) , where θc are control policy parameters , such as a feedback matrix for LQR . However , in practice , given a possibly noisy forecast ŝt : t+H−1 , it will enact a control denoted by ût = π ( xt , ŝt : t+H−1 ; θc ) , which implicitly depends on the encoder/decoder parameters θe , θd via the forecast ŝ . Control Cost : Our main objective is to minimize end-to-end control cost Jc , which depends on initial state x0 and controls û0 : T−1 , which in turn depend on the forecast ŝ0 : T−1 . For a simpler notation , we use bold variables to define the full timeseries , i.e. , u : = u0 : T−1 , s : = s0 : T−1 , û : = û0 : T−1 and ŝ : = ŝ0 : T−1 . The control cost Jc is a sum of stage costs c ( xt , ût ) and terminal cost cT ( xT ) : Jc ( û ; x0 , s ) = cT ( xT ) + ∑T−1 t=0 c ( xt , ût ) , where xt+1 = f ( xt , ût , st ) , t ∈ { 0 , · · · , T −1 } . Importantly , the above plant dynamics f evolve according to true timeseries st , but controls ût are enacted with possibly noisy forecasts ŝt . Forecasting Errors : In practice , a designer often wants to visualize decoded forecasts ŝ to debug anomalies or view trends . While our principal goal is to minimize the control errors and cost associated with forecast ŝ , we allow a designer to optionally penalize mean squared prediction error ( MSE ) . This penalty incentivizes a forecast ŝt to estimate the key trends of st , serving as a regularization term : JF ( s , ŝ ) = 1T ∑T−1 t=0 ||st − ŝt||22 . Overall Weighted Cost : Given our principal objective of minimizing control cost and optionally penalizing prediction error , we combine the two costs using a user-specified weight λF . Importantly , we try to minimize the additional control cost Jc ( û ; x0 , s ) incurred by using forecast ŝ instead of true timeseries s , yielding overall cost : J tot . ( u , û , s , ŝ ; x0 , λ F ) = 1 T ( Jc ( û ; x0 , s ) − Jc ( u ; x0 , s ) ︸ ︷︷ ︸ extra control cost ) + λFJF ( s , ŝ ) . ( 1 ) The total cost implicitly depends on controller , encoder , and decoder parameters via controls u and û and the forecast ŝ . 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 a controller π ( ; θc ) with fixed , pre-trained parameters θc , fixed bottleneck dimension Z , and perfect measurements of internal controller state x0 : T . Given a true exogenous timeseries s0 : T−1 drawn from data distribution D , find encoder and decoder parameters θe , θd to minimize the weighted control and forecasting cost ( Eq . 1 ) with weight λF : θ∗e , θ ∗ d = argmin θe , θd Es0 : T−1∼D [ J tot . ( u , û , s , ŝ ; x0 , λF ) ] , where φt = gencode ( st−W+1 : t ; θe ) , φt ∈ RZ ŝt : t+H−1 = gdecode ( φt ; θd ) , ût = π ( xt , ŝt : t+H−1 ; θc ) , ut = π ( xt , st : t+H−1 ; θc ) , xt+1 = f ( xt , ût , st ) , and xt ∈ X , ût ∈ U , t ∈ { 0 , · · · , T − 1 } . Technical Novelty and Practicality of our Co-design Problem : Having formalized our problem , we can now articulate how it differs from classical networked control and tele-operation [ 18–20 , 6 , 5 ] , compressed sensing [ 21 , 22 ] , and certainty-equivalent control [ 23 , 24 ] . 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 as the Kalman Filter [ 26 ] , and a “ certainty-equivalent ” controller ( the linear quadratic regulator ) for optimal performance . This is because the timeseries owner measures a non-stationary timeseries st ( 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 to equally weighting and sending the full ŝt , which a classic state observer in LQG would do . Moreover , even when the estimator and controller are separated by a bandwidth-limited network and the separation principle does not hold [ 27 ] , our setting still differs from classical networked control [ 18–20 , 6 , 5 ] . These works assume that both the full plant state xt and controls ut are encoded and transmitted between a remote controller and plant . In stark contrast , the only transmitted data in our setting is external information st from a network operator , which can improve an independent controller ’ s local decisions ut based on its internal state xt . As such , simply grouping controller state xt and network timeseries st into a joint state for classical tele-operation is infeasible , since xt and st are measured at different locations by different entities . In essence , Prob . 1 formalizes how a network operator can provide significant value to an independent controller by judicious data sharing .

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.