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 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.