Prediction, Consistency, Curvature: Representation Learning for Locally-Linear Control
Authors: Nir Levine
Yinlam Chow
Rui Shu
Ang Li
Mohammad Ghavamzadeh
Hung Bui
Venue:  (2020)
Keywords: Embed-to-Control
Representation Learning
Stochastic Optimal Control
VAE
iLQR

Abstract: Many real-world sequential decision-making problems can be formulated as optimal control with high-dimensional observations and unknown dynamics. A promising approach is to embed the high-dimensional observations into a lower-dimensional latent representation space, estimate the latent dynamics model, then utilize this model for control in the latent space. An important open question is how to learn a representation that is amenable to existing control algorithms? In this paper, we focus on learning representations for locally-linear control algorithms, such as iterative LQR (iLQR). By formulating and analyzing the representation learning problem from an optimal control perspective, we establish three underlying principles that the learned representation should comprise: 1) accurate prediction in the observation space, 2) consistency between latent and observation space dynamics, and 3) low curvature in the latent space transitions. These principles naturally correspond to a loss function that consists of three terms: prediction, consistency, and curvature (PCC). Crucially, to make PCC tractable, we derive an amortized variational bound for the PCC loss function. Extensive experiments on benchmark domains demonstrate that the new variational-PCC learning algorithm benefits from significantly more stable and reproducible training, and leads to superior control performance.  Further ablation studies give support to the importance of all three PCC components for learning a good latent space for control.

Published as a conference paper at ICLR 2020
PREDICTION, CONSISTENCY, CURVATURE: REPRESEN- TATION LEARNING FOR LOCALLY-LINEAR CONTROL
Nir Levine'; Yinlam Chow?", Rui Shu Ang Li', Mohammad Ghavamzadeh"s Hung Bui 1 DDepMindd "Google Research, "Stanford University. "Facebook Al Research, VinAl
ABSTRACT
Many real-world sequential decision-maiee makinng problems can be formulated as op- timal control with high-dimensional observations and unknown dynamics. A promising approach is to embed the high-dimensional observations into a lower- dimensional latent representation space, estimate the latent dynamics model, then utilize this model for control in the latent space. An important open question is how to learn a representation that is amenable to existing control algorithms? In this pa- per, we focus on learning representations for locally-linear control algorithms, such iterative LQR (iLQR). By formulating and analyzing the representation learning problem from an optimal control perspective, we establish three underlying princi- ples that the learned representation should comprise: 1) accurate prediction in the observation space, 2) consistency between latent and observation space dynamics, and 3) low curvature in the latent space transitions. These principles naturally correspond to a loss function that consists of three terms: prediction, consistency, and curvature (PCC). Crucially, to make PCC tractable, we derive an amortized variational bound for the PCC loss function. Extensive experiments benchmark domains demonstrate that the new variational-PCC learning algorithm benefits from significantly more stable and reproducible training, and leads to superior control performance. Further ablation studies give support to the importance of all three PCC components for learning a good latent space for control.
INTRODUCTION
Decomposing the problem of drcision-making in an unknown environment into estimating dynamics followed by planning provides powerful framework for building intelligent agents. This decomposi- tion confers several notable benefits. First, enables the handling of sparse-reward environments by leveraging the dense signal of dynamics prediction. Second, once a dynamics model is learned, it can be shared across multiple tasks within the same environment. While the merits of this decomposition have been demonstrated in low-dimensional environments (Deisenroth & Rasmussen, 2011; Gal et al., 2016), scaling these methods to high-dimensional environments remains an open challenge.
The recent advancements in generative models have enabled the successful dynamics estimation of high-dimensional decision processes (Watter et al., 2015; Ha & Schmidhuber, 2018; 8; Kurutach et al., 2018). This procedure of learning dynamics can then be used in conjunction with a plethora of decision-maker techniques, ranging from optimal control to reinforcement learning (RL) (Watter et al.. 2015; Banijamali et al., 2018; Finn et al., 2016: Chua et al., 2018; Ha & Schmidhuber, 2018; Kaiser al., 2019; Hafner al., 2018; Zhang et al., 2019). One particularly promising line of work in this area focuses on learning the dynamies and conducting control in a low-dimensional latent embedding of the observation space, where the embedding itself is learned through this process (Watter et al., 2015; Banijamali et al., 2018; Hafner al., 2018; Zhang et al., 2019). We refer to this approach as learning controllable embedding (LCE). There have been two main approaches to this problem: 1) to start by defining a cost function in the dimeesiimnsional observation space and learn the embedding space, its dynamics, and reward function, by interacting with the environment in RL fashion (Hafner et al., 2018: Zhang et al., 2019), and 2) first learn the embedding space and its dynamics, and then define a cost function in this oowdiimesiional space and conduct the control ( (Watter et al., 2015; Banijamali et al., 2018). This can be later combined with RL for extra fine-tuning of the model and control.
In this paper, we take the second approach and particularly focus on the important question of what desirable traits should the latent embedding exhibit for it to be amenable to a specific class of control/learingg algorithms, namely the widely used class of locally-lineat -inear control (LLC) algorithms? We argue from an optimal control standpoint that our latent space should exhibit three properties. The first is prediction: given the ability to encode to and decode from the latent space, we expect
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Published as a conference paper at ICLR 2020
the process of encoding, transitioning via the latent dynamics, and then decoding, to adhere to the true observation dynamics. The second is consistency: given the ability to encode observation trajectory sampled from the true environment, we expect the latent dynamics to be consistent with the encoded trajectory. Finally, curvature: in order to learn a latent space that is specifically amenable to LLC algorithms, we expect the (learned) latent dynamics to exhibit low curvature in order to minimize the approximation error of its first-order Taylor expansion employed by LLC algorithms. Our contributions are thus as follows: (1) We propose the Prediction, Consistency, and Curvature (PCC) framework for learning latent space that is amenable to LLC algorithms and show that the elements of PCC arise systematically from bounding the suboptimality of the solution of the LLC algorithm in the latent space. (2) We design a latent variable model that adheres to the PCC framework and derive a tractable variational bound for training the model. (3) To the best of our knowledge, our proposed curvature loss for the transition dynamics (in the latent space) novel. We also propose a direct amortization of the Jacobian calculation in the curvature loss to help training with curvature loss more efficiently. (4) Through extensive experimental comparison, we show that the PCC model consistently outperforms E2C (Watter et al., 2015) and RCE (Banijamali et al., 2018) on a number of centrol-fromimman model to have consistency and low-curvature.
tasks, and verify via ablation, the importance of regularizing the
2 PROBLEM FORMULATION
We are interested in controlling the non-linear dynamical systems of the form S1+1 = fst ue) + w. over the horizon T. In this definition, St E S C IR" and u, E u C R" are the state and action of the system at time step E :O.....I T 13. w the Gaussian system noise, and fs a smooth non-linear system dynamics. We are particularly interested in the scenario in which we only have access to the high-dimensional observation I, E X' C RT of each state 8, (Na >> nt).. This scenario has application in many real-world problems, such as visual-servoing (Espiau et al., 1992), in which we only observe high-dimensional images of the environment and not its underlying state. We further assume that the high- nigh-dimensional observations x have been selected such that for any arbitrary control sequence U = { the observation sequence {x² T Jt= is generated by stationary Markov process, i.e., It+1 ~ P(-|2a, uz), Ut), Vt E (o...., 1}.
A common approach to control the above dynamical system is to solve the following stochastic optimal control (SOC) problem (Shapiro et al., 2009) that minimizes expected cumulative cost:
min L,U, zo) == B(or - Σ ca ux) P.
(SOC1)
1-0
where c;: X'' x4 - R>u is the immediate cost function at time t, cr E Rz is the terminal cost, and Ty is the observation at the initial state so- Note that all immediate costs are defined in the observation space X, and are bounded by Cmax > 0 and Lipschitz with constant Cip > 0. For example, in visual- servoing, (SOC1) can be formulated as a goal tracking problem (Ebert et al., 2018), where we control the robot to reach the goal observation Tgoal. and the objective is to compute sequence of optimal open-loop actions U that minimizes the cumulative tracking error EE E, ||= - x =g 1| P. P.)))
Since the observations æ are high dimensional and the dynamics in the observation space P(- uz) is unknown, solving (SOCI) is often intractable. To address this issue, a class of algorithms has been recently developed that is based on learning low-dimensional latent (embedding) space Z C R (nz < n₂)) and latent state dynamics, and performing optimal control there. This class that we refer to as learning controllable embedding (L.CE) throughout the paper, include recently developed algorithms, such as F2C (Watter et al., 2015), RCE (Banijamali et al., 2018) 8), and SOLAR (Zhang et al., 2019). The main idea behind the LCE approach is to learn triplet. (i) an encoder : X - (Z); (ii) a dynamics in the latent space F: Z x l + P(Z): )); and (iii) a decoder D : + P(X). These in turn can be thought of as defining a (stochastic) mapping P X' xu + P(X') of the form P = D o F Fold E. We then wish to solve the SOC in latent space Z:
min U.P
E LUU F,c, 20) E, E,20 + X R2(P),
(SOC2)
such that the solution of (SOC2), Ug. has similar performance to that of (SOCI), Ui. ie., L(Ur, P. c, 20) x L(U c, c.). In (SOC2), 20 is the initial latent state sampled from the encoder E(-120); ē : 2 U + R20 is the latent cost function defined as d uz) =
I ce(xz, un (D) (2( is a regularizer over the mapping P; and X2 is the corresponding
A method to ensure this Markovian assumption is by butfering observations (Mnih et al., 2013) for number of time steps.
See Appendix B.3 for the extension to the closed-loop MDP problem.
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E
B
JO
F
20
F1
P
P
Figure I: Evolution of the states (a)(blue) in equation SOCI under dynamics P. (b)(green) in equa- tion SOC2 under dynamics F, and (c)(rd) in equation SOC3 under dynamics P.
regularization parameter. We will define R2 and A2 more precisely in Section 3. Note that the expectation in (SOC2) is over the randomness generated by the (stochastic) encoder E.
3 PCC MODEL: A CONTROL PERSPECTIVE
As described in Section 2, we are primarily interested in solving (SOC1), whose states evolve under dynamics P. as shown at the bottom row of Figure (() in (blue). However, because of the difficulties in solving (SOC1), mainly due to the high dimension of observations x, LCE proposes to learn mapping P by solving (SOC2) that consists of a loss function, whose states evolve under dynamics F (after an initial transition by encoder E), as depicted in Figure 1(b), and regularization term. The role of the regularizer R2 is to account for the performance gap between (SOC1) and the loss function of (SOC2), due to the discrepancy between their evolution paths, shown in Figures ((a)(blue)
and (b)(gre))). The goal of LCE is to learn P of the particular form P D F a E., described in Section 2, such that the solution of (SOC2) has similar performance to that of (SOC1). In this section, we propose a principled way to select the regularizer R2 to achieve this goal. Since the exact form of (SOC2) has direct effect on learning P. designing this regularization term, in turn, provides us with recipe (loss function) to learn the latent (embedded) space Z. In the following subsections, show that this loss function consists of three terms that correspond to prediction, consistency, and curvature, the three ingredients of our PCC model.
Note that these two SOCs evolve in two different spaces, one in the observation space X' under dynamics P. and the other one in the latent space Z (after an initial transition from X' 2) under dynamies F. Unlike P and that only operate in a single space, X' and Z, respectively, P can govern the evolution of the system in both 2' and (see Figure (c))). Therefore, any recipe to learn P. and as result the latent space 2, should have at least two terms, to guarantee that the evolution paths resulted from P in X' and Z are consistent with those generated by P and F. We derive these two terms, that are the prediction and consistency terms the loss function used by our PCC model, in Sections 3. and 3.2, respectively. While these two terms are the result of learning e in general SOC problems, in Section 3.3, we concentrate on the particular class of LLC algorithms (e.g.. iLQR (Li & Todorov, 2004)) to solve SOC, and add the third term, curvature, to our recipe for learning P.
3.1 PREDICTION OF THE NEXT OBSERVATION
Figures ((a)(blue) and (re) show the transition in the observation space under P and P. where Ie is the current observation, and T1+1 and &t-1 are the next observations under these two dynamics, respectively. Instead of learning a P with minimum mismatch with P in terms of some distribution norm, we propose to learn P by solving the following SOC:
min U.P
L(U, P, ro) + Asy (P()),
(SOC3)
whose loss function is the same as the one in (SOC1), with the true dynamics replaced by P. In Lemma I (see Appendix A.1, for proof), we show how to set the regularization term Rs in (SOC3), such that the control sequence resulted from solving (SOC3), Už, has similar performance to the solution of (SOC1), Uf, i.e., L(U P, e, xo) As L(U C, 20).
Lemma 1. Let U be a solution to (SOCI) and (Uz, PT) be a solution to (SOC3) with R₂ ()) = EE,, Dxu (P(- x)]||((|xz; u and
Ag = 2 T Cmax-
Then, we have L(U 20) Z LUU,, - 2 Rs P)
In Eq. 1, the expectation is over the state-action stationary distribution of the policy used to generate the training samples (uniformly random policy in this work), and U is the Lebesgue measure of U.
In the case when sampling policy is non-uniform and has no measure-zero set, 1/U its minimum measure.
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3.2 CONSISTENCY IN PREDICTION OF THE NEXT LATENT STATE
In Section 3.1, provided a recipe for learning P (in form of D o Fo E) by introducing an intermediate (SOC3) that evolves in the observation space X' according to dynamics P. In this section we first connect (SOC2) that operates in Z with (SOC3) that operates in X. For simplicity and without loss generality, assume the initial cost co (x, u) is zero.* 4 Lemma (see Appendix A.2, for proof) suggests how we shall set the regularizer in (SOC2), such that its solution performs similarly to that of (SOC3), under their corresponding dynamics models.
Lemma 2. Let (U3, P3) be a solution to (SOC3) and (U2, P2) be a solution to (SOC2) with
R2(P) = BRr, DKK (R P) (-|r,, w)l(( o (11;,))))
and x2 v 2U T Cmax- (2)
Then, we have L(U3, P3 C, ro) 2 L(U Pj. c, 2o) 2\2 R3
Similar to Lemma 1, in Eq. 2, the expectation is over the state-action stationary distribution of the policy used to generate the training samples. Moreover, (E P) (='1z, u) S. F(='\==')dP(C )dP(x'|an, u) and (F (2'12z, u) Jg F(z'|2, u)dE(z) are the probability over the next latent state 2', given the current observation x and action u, in (SOC2) and (SOC3) (see the paths T2 + + ăt+1 and Je - 2, + žt+1 + 1c+1 2++1 in Figures ((b)gre) and ((c)(rd)). Therefore (P) can be interpreted as the measure of discrepancy between these models, which we term as consistency loss.
Although Lemma provides a recipe to learn P by solving (SOC2) with the regularizer (2), unfortu- nately this regularizer cannot be computed from the data - that is of the form (It, Wr, Fr++1 - because the first term in the Dxi. requires marginalizing over current and next latent states (2r and žt-1 1 in Figure (())). To address this issue, we propose use the (computable) regularizer
R (P) = E E.,,, DKL E(+|=") o (1
(3)
in which the expectation is over (z, u, x²) sampled from the training data. Corollary (see Appendix A.3, for proof) bounds the performance loss resulted from using (P) instead of (P), and shows that it could be still reasonable choice.
Corollary 1. Let (Uz, PT) be a solution to (SOC3) and (Uz, P2) be a solution to (SOC2) with (P) and and X2 defined by (3) and (2). Then, we have L(Už, P3. co) 2 L(U2, Pr c.
2 2R 2Rg(()).
Lemma suggests a regularizer Rg to connect the solutions of (SOC1) and (SOC3). Similarly, Corollary 1 shows that regularizer R in (3) establishes connection between the solutions of (SOC3) and (SOC2). Putting these results together, we achieve our goal in Lemma 3 (see Appendix A.4, for proof) design a regularizer for (SOC2), such that its solution performs similarly to that of (SOCI).
Lemma 3. Let U. be a solution to (SOCI) and (Uz, PT) be a solution to (SOC2) with
(P( = 3Rs + 2R (P)
and A2 = - Cmax:
(4)
where (P) and (P) are defined by (1) and (3). Then, we have
L(U1, P, P,C, 2 L(U2 P,,, 2dd V R2(P
3.3 LOCALLY- CONTROL IN THE LATENT SPACE AND CURVATURE REGULARIZATION
In Sections 3.1 and 3.2, we derived a loss function to learn the latent space Z. This loss function, that was motivated by the general SOC perspective, consists of two terms to enforce the latent space to not only predict the next observations accurately, but to be suitable for control. In this section, we focus on the class of locally-linear control (LLC) algorithms (e.g., iLQR), for solving (SOC2), and show how this choice adds third term, that corresponds to curvature, to the regularizer of (SOC2), and as a result, to the loss function of our PCC model.
The main idea in LLC algorithms is to iteratively compute an action sequence to improve the current trajectory, by linearizing the dynamics around this trajectory, and use this action sequence to generate
"With non-zero initial cost, similar results can he derived by having an additional consistency term on zig-
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the next trajectory (see Appendix B for more details about LLC and iLQR). This procedure implicitly assumes that the dynamics is approximately locally linear. To ensure this in (SOC2), we further restrict the dynamics P and assume that it is not only of the form P = D o F o E, but F, the latent space dynamics, has low curvature. One way to ensure this in (SOC2) is to directly impose penalty over the curvature of the latent space transition function fz (z, u). Assume F(z, u) fz u) + w. where w a Gaussian noise. Consider the following SOC problem:
min U.P
E [L, F,), Fo) + / (2(P) + Patc(P),
(SOC-LLC)
where R2 is defined by (4); U is optimized by LLC algorithm, such iLQR; REIC( (P) is given by,
Buce (P) = Bn,u [B. [f + ex, u + cw) - f2(2, u) (000)-(V_fffeee (V=f2(z,u) u) + Vufz(z, u) e. ) E (5)
where = (cz, (N00 1), 5 > O is a tunable parameter that characterizes the "di- ameter" of latent state-action space in which the latent dynamics model has low curvature.
where 1/X is the minimum non-zero measure of the sample distribution w.r.t. X, and 1 - n E [0, 1) is a probability threshold. Lemma 4 (see Appendix A.5, for proof and discussions on how 5 affects LLC performance) shows that a solution of (SOC-LLC) has similar performance to solution of (SOCI, and thus, (SOC-LLC)
ALL = 2 VT max (nip(1 + (22lo((T/)))
a reasonable optimization problem to learn P. and also the latent space Z.
Lemma 4. Let (Unee Pite) be a LLC solution to (SOC-LLC) and Uj be a solution to (SOCI). Suppose the nominal latent state-action trajectory {(z,, u) Yo satisfies the condition: (Zz, uz) ~ ((2,2 u2,1), 821), where {(22,01 (2,0) TT- is the optimal trajectory of (SOC2). Then with proba-
bility 1 - we have L(U;, c, 20) 2 L(Uūc: P,c, mo) 22LL Re(Pitc) + RRLLC (Pic)
In practice, instead of solving (SOC-LLC) jointly for and P. we treat (SOC-LLC) as a bi-level optimization problem, first, solve the inner optimization problem for P, i.e.
p- E arg min
(P) + de (P) (ecr
(PCC-LOSS)
P
where (P)(P) = - (log ((zl|z,, u)] is the negative log-likelihood," and then, solve the outer optimization problem, ming L(U, F C, 20), where p= nr f , to obtain the optimal control sequence U Solving (SOC-LLC) this way is an approximation, in general, but is justified, when the regularization parameter ALLC is large. Note that we leave the regularization parameters ( (Ap. Ae, Acur) as byper-parameters of our algorithm, and do not use those derived in the lemmas of this section. Since the loss for learning f in (PCC-LOSS) enforces (i) prediction accuracy, (ii) consistency in latent state prediction, and (iii) low curvature over fz, through the regularizers R/z, Ryj, and RLLC. respectively, we refer to it as the prediction-comsistencyy curvvature (PCC) loss.
4 INSTANTIATING THE PCC MODEL IN PRACTICE
The PCC-Model objective in (PCC-LOSS) introduces the optimization problem minp (P) + (P) + (ewr To instantiate this model in practice, we de- scribe p= D Fo E as a latent variable model that factorizes as P Ze, & Ze. u.) P(z P()+1 | 2t. u.) P( +1 1.41). In this section, we propose a variational approximation to the intractable negative log-likelihood R2 and batch-consistency R losses, and an efficient approximation of the curvature loss RLLC-
4.1 VARIATIONAL PCC
The negative log-likoo likelhooooo R2 admits a variational bound via Jensen's 's Incquality,
P(x Zt, Bett we. Us)
(P) = log P(ret1 [r, us) = log Eq(z (e
Q(2e, Bet1 De, Me, ze+1)
Plect1, Ze, Set1
Ze. (e)
< -B
log
= Rou (PQ)),
(6)
Q(ze, Bets Te, Me, Fe+1)
SSince Ra (P) is the sum of R' (P) and the entropy of P, we replaced it with (P(P) in (PCC-LOSS)).
Forr notation convenience, drop the expectation over the empirical data that appears various loss terms.
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which holds for any choice of recognition model Q. For simplicity, we assume the recog- nition model employs bottom-up inference and thus factorizes as Q(z, & Tx+1, ue) @(2+1|/T+41 (FF+4)Q((z)) [=(+=[[++=+=++>)+0 Lt, uz). The main idea behind choosing a mackward-facing model is to allow the model to learn to account for noise in the underlying dynamics. We estimate the expec- tations in (6) via Monte Carlo simulation. To reduce the variance of the estimator, we decompose Bo, further into
[log P(ect 12441) + (Q((e)) D (Q(z) 841, u.)l mi)]
-Eq(Ae+1 (xc-1)
-H (Q( m=+1)) E Q(844118441)
log P(eet1 11 ze, u.]
and note that the Entropy (-) and Kullback-Leibler (-I.- terms are analytically tractable when Q is restricted to a suitably chosen variational family (i.e. in our experiments, Q( (++1) and Q(= Ec+1; Te, U²) are factorized Gaussians). The derivation is provided in Appendix C.I.
Interestingly, the consistency loss Ri admits a similar treatment. We note that the consisteney loss seeks to match the distribution of 2++1 Øt, Ut with 24+1 T441, which we represent below as
(P) = DKL (P(a (ee1)) It, ui) - (P(z xe+1)) - E B ( log P( 22, uz)
#241
Here, P(e+41 It, ue) is intractable due the marginalization of zt. We employ the same procedure as in (6) to construct tractable variational bound
P(z,, 2--1 xz, We)
(P) S (P(ea)) ((41))) E Pizee ( Equa & 1.Pa.
log Q(za 2141, w.)
We now make the further simplifying assumption that Q(a++1 | xo+1) P(ec+1 x241). This allows us to rewrite the expression as
(P) S - H(Q((++1 m++1)) - Q(4++1l944)) [log P(eet1 2e, ue) Q(arkee4
B 11 (Q(a. 8141, u/) P((z mx))] = R2 Romad (P.Q).
which is a subset of the terms in (6). See Appendix C.2 for a detailed derivation.
4.2 CURVATURE REGULARIZATION AND AMORTIZED GRADIENT
variant of the curvature loss where Taylor expansions and gradients are evaluated at + E2 and ū = u + Ex.
In practice
RLLC Eo 1If222, (V=ff(,,, + Vafz(2,
- f2(2, x)
(8)
When n, is large, evaluation and differentiating through the Jacobians can be slow. To circumvent this issue, the Jacobians evaluation can be amortized by treating the Jacobians as the coefficients of the best linear approximation at the evaluation point. This leads to a new amortized curvature loss
RLoc-Amor (P, P, B) = (E- fzl u) (A(z, T)e. + B(z, ū)e.. fz(z, u))
(9) be optimized. Intuitively, the amortized curvature loss
where A and B are function approximators seeks-for any given (z, to find the best choice of linear approximation induced by A(z, u) and B(z, u) such that the behavior of F2 in the neighborhood of (z, u) is approximately linear.
5 RELATION TO PREVIOUS EMBED-TO-CONTROO
APPROACHES
In this section, we highlight the key differences between PCC and the closest previous works, namely E2C and RCE. A key distinguishing factor is PCC's 's use of nonlinear latent dynamics model paired with an explicit curvature loss. In comparison, E2C and RCE both employed "locally-linear dynamics" of the form to perturbations of z and u. When contrasted with (9), it is clear that neither and B in the E2C/RCE formulation can be treated as the Jacobians of the dynamics, and hence the curvature of the dynamics is not being controlled explicitly. Furthermore. since the locally-linear dynamics are wrapped inside the maximum maximum-likelihood estimation, both E2C and RCE conflate the two key elements prediction and curvature together. This makes controlling the stability of training much more difficult. Not only does PCC explicitly separate these two components, we are also the first to explicitly demonstrate theoretically and empirically that the curvature loss is important for il QRR.
A(z, 7)= 2 + B(z, m)u c(2, ū) where 2 and ū auxiliary random variables meant

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Figure 2: Top: Planar latent representations; Bottom: Inverted Pendulum latent representations (randomly selected); left two: RCE, middle E2C, right two: PCC.
Furthermore, RCE does not incorporate PCC's consistency loss. Note that PCC, RCE, and E2C all Markovian on minimizing the prediction loss will result in a discrepancy between how the model is trained wranggg-raaiiiiinn-ddeeeee
frameworks. Under such a framework, the sole reliance
(maximizing the likclihood induced by test-time for control (continual transitioning the latent space without ever decoding). By explicitly minimizing the consistency loss, PCC reduces the discrapancy between how the model is trained versus how it is used at test-time for planning. Interestingly, E2C does include regularization term that is akin to PCC's consistency loss. However, as noted by the authors of RCE, E2C's maximization of maiiina log- lieliiiiiihooss of (Z,, It+1) as opposed to the conditional likelihood of x++1 given Ie means that E2C does not properly minimize the prediction loss prescribed by the PCC framework.
versus how it is used at
6 EXPERIMENTS
In this section, we compare the performance of PCC with two model-based control algorithm baselines: RCE" (Banijamali et al., 2018) and E2C (Watter et al., 2015), well running thorough ablation study on various components of PCC. The experiments are based on the following continuous control benchmark domains (see Appendix D for more descriptions): (i) Planar System, (ii) Inverted Pendulum, (iii) Cartpole, (iv) 3-link manipulator, and (v) TORCS simulator (Wymann et al., 2000).
To generate our training and test sets, each consists of triples (I2, ue Ft+1). we: (1) sample an under- lying state 8 and generate its corresponding observation 2t, (2) sample an action u4, and (3) obtain the next state 8++1 according to the state transition dynamics, add it zero-mean Gaussian noise with variance ,²I,,- and generate corresponding observation 11+1. To ensure that the observation-action data is uniformly distributed (see Section 3), we sample the state-action pair (St: 14) uniformly from the stateaaction space. To understand the robustness of each model, we consider both deterministic (a = 0) and stochastic scenarios. In the stochastic case, we add noise to the system with different values of and evaluate the models' performance under various degree of noise.
Each task has underlying start and goal states that unobservable to the algorithms, instead, the algorithms have access to the corresponding start and goal observations. We apply control using the iLQR algorithm (see Appendix B), with the same cost function that was used by RCE and E2C, namely, ō(2,, u4) = (2+ Zgoal T Q(4 Zgou) + u, Rug, and c(2)) = (2r Zgoal) Q(2r Zgow), where Zgeat is obtained by encoding the goal observation. and Q = K In,, R = In. Details of our implementations are specified in Appendix D.3. We report performance in the underlying system, specifically the percentage of time spent in the goal region"". 10
A Reproducible Experimental Pipeline In order to measure performance reproducibility, we perform the following 2-step pipeline. For each control task and algorithm, we (1) train 10 models
For the RCE implementation, we directly optimize ELBO loss in Equation (16) of the paper. We also tried the approach reported in the paper on increasing the weights of the two middle terms then annealing them to 1. However, in practice this method is sensitive to annealing schedule and has convergence issues. wew.oouttu.e/GrgAAA....
Sec a control demo on the TORCS simulator at
AAcordinn to the definition of latent cost ह(2, u) Do c(z, u), its quadratic approximation is given by
a(2, (z.u) w)
Zgnaa u
Docl
2
Zgeal
Doc 200
2
Zgool
V.J
us
Yet for simplicity, we choose the same latent cost as in RCE and E2C with fixed, tunable matrices Q and R. Another possible metric is the average distance to goal, which has similar behavior.
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Table 1: Percentage of steps in goal state. Averaged over all models (left), and the best model (right).
Domain
RCE (all)
E2C (all)
PCC (all)
RCE (top 1)
E2C (top 1)
PCC (top 1)
Planar
2,1 1 0.8
5,5
1.7
35.7 1 3,4
9.2 1 1.4
36.5 + 3.6
72.1 1 0.4
Pendulum
24.7 = 3.1
46.8 1 4.1
58.7 1 3.7
68.8
1: 2.2
89.7 1: 0.5
90.3 1 0.4
Caripole
59,5 4.1
1.311
54.3 3.9
99,45 1 0.1
40.2 3.2
93.9 1.7
3-link
1.1 1
0.4
4.7 1
1.1
18.8 1
2.1
10.6
1 0.8
20.9
1 0.8
47.2 1 1.7
TORCS
27.4 1.8
28.2
1 1.9
60.7 1 1.1
39.9 2.2
1
54.1 1 2.3
68.6 1 0.4
Table 2: Ablation analysis. Percentage of steps spent in goal state. From left to right: PCC including all loss terms, excluding consistency loss, excluding curvature loss, amortizing the curvature loss.
Domain
PCC
PCC no Con
PCC no Cur
PCC Amor
Planar
35.7
1 3.4
0.0 1 0.0
29.6 1 3.5
41.7 1 3.7
Pendulum
58.7
1: 3.7
52.3 1 3.5
50.3 + 3.3
54.2 1 3.1
Cartpole
54.3
3,9
5.1 + 0.4
17.4 1.6
14.3 + 1.2
3-link
18.8 2.1
9.1 + 1.5
13.1 1.9
I1.3 U..EI1...
independently, and (2) solve 10 control tasks per model (we do not cherry-pick, but instead perform
a total of 10 x 10 = 100 control tasks). We report statistics averaged over all the tasks (in addition, we report the best performing model averaged over its 10 tasks). By adopting principled and statistically reliable evaluation pipeline, also address pitfall of the compared baselines where the best model needs to be cherry picked, and training variance was not reported.
Results Table 1 shows how PCC outperforms the baseline algorithms in the noiseless dynamics case by comparing means and standard deviations of the means on the diffferent control tasks (for the case of added noise the dynamics, which exhibits similar behavior, refer to Appendix E.1). It is important to note that for each algorithm, the performance metric averaged over all models is drastically different than that of the best model, which justifies our rationale behind using the reproducible evaluation pipeline and avoid cherry-picking when reporting. Figure 2 depicts 2 instances (randomly chosen from the 10 trained models) of the learned latent space representations on the noiseless dynamics of Planar and Inverted Pendulum tasks for PCC, RCE, and E2C models (additional representations can be found in Appendix E.2). Representations were generated by encoding observations corresponding to a uniform grid over the state space. Generally, PCC has a more interpretable representation of both Planar and Inverted Pendulum Systems than other baselines for both the noiseless dynamics case and the noisy case. Finally, in terms of computation, PCC demonstrates faster training with 64% improvement over RCE, and 2% improvement over E2C 11
Ablation Analysis On top of comparing the performance of PCC to the baselines, in order to understand the importance of each component in (PCC-LOSS), we also perform an ablation analysis on the consistency loss (with/without consistency loss) and the curvature loss (with/without curvature loss, and with/without amortization of the Jacobian terms). Table 2 shows the ablation analysis of PCC on the aforementioned tasks. From the numerical results, one can clearly see that when consistency loss is omitted, the control performance degrades. This corroborates with the theoretical results in Section 3.2, which indicates the relationship of the consistency loss and the estimation error between the next-latent dynamics prediction and the next-latent encoding. This further implies that as the consistency term vanishes, the gap between control objective function and the model training loss is widened, due to the accumulation of state estimation error. The control performance also decreases when one removes the curvature loss. This is mainly attributed to the error between the iLQQ control algorithm and (SOC2). Although the latent state dynamics model is parameterized with neural networks, which are smooth, without enforcing the curvature loss term the norm of the Hessian (curvature) might still be high. This also confirms with the analysis in Section 3.3 about sub-optimality performance and curvature of latent dynamics. Finally, we observe that the performance of models trained without amortized curvature loss are slightly better than with their amortized counterpart, however, since the amortized curvature loss does not require computing gradient of the latent dynamics (which means that in stochastic optimization one does not need to observe relative speed-ups in model training with the amortized version
estimate its Hessian), (speed-up of 6%, 9%, and 15% for Planar System, Inverted Pendulum, and Cartpole, respectively).
7 CONCLUSION
In this paper, we argue from first principles that learning a latent representation for control should be guided by good prediction in the observation space and consistency between latent transition and
II Comparison jobs were deployed on the Planar system using Nvidia TITAN Xp GPU.
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the embedded observations. Furthermore, if variants of iterative LQR used the controller, the low-curvature dynamics desirable. All three elements of our PCC models are critical the stability of model training and the performance of the in-latent-space controller. We hypothesize that each particular choice of controller will exert different requirement for the learned dynamics. A future direction is identify and investigate the additional bias for learning an effective embedding and latent dynamics for other type of model-based control and planning methods.
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A TECHNICAL PROOFS OF SECTION 3
A. PROOF OF LEMMA 1
Following analogous derivations of Lemma 11 in Petrik et al. (2016) (for the of finite-horizon MDPs), for the case of finite-horizon MDPs, one has the following chain of inequalities for any given control sequence ( and initial observation zg:
LLU, P. 20) - -LU,P,
E
cT (T(
( 46) P,
E
cr (2T) Σ
(12,, ue) P,200
2T Cmax
(rv(P(1
u.)l ut)) x0
T-1
<V2T2 Cmax E
T
1=0
<V2T2 Cmax
E
ul 20
T
where Drv is the total variation distance of two distributions. The first inequality is based on the result of the above lemma, the second inequality is based on Pinsker's inequality (Ordentlich & Weinberger, 2005), and the third inequality is based on Jensen's 's inequality (Boyd Vandenberghe, 2004) of (+) function.
Now consider the expected cumulative KL cost: E ITTO KL(P(- u.)l u.)) z00
with respect to some arbitrary control action sequence {ut]T=] Notice that this arbitrary action sequence can always be expressed in form of deterministic policy u, = r'' r' t) with some non- stationary state-actis action mapping n'. Therefore, this KL cost be written as:
KL(((11xx, u.)l1P(((ar, uz)) x, 2o
=F
(L((P()pp,, ux)l ue )dd (oddrr(ueeel (weker 1) ())
P,
(10)
dr' (uel(e))) dU P. dU(v))
E
KL(P(- ue) (1
<U Em,,
(1)
where the expectation is taken over the state-action occupation measure $ ΣT P(x, 2, ue To, U' of the finite-horizon problem that is induced by data- sampl--g policy U. The last inequality due to change of measures in policy, and the last inequality is due the facts that (i) is deterministic policy, (ii) dU (we) is sampling policy with lebesgue measure 1/U over all control dx((a., - d(((,))
actions, (iii) the following bounds for importance sampling factor holds:
To conclude the first part of the proof, combining all the above arguments we have the following inequality for any model P and control sequence U:
L(U, L(U,P, P, L(U, L(U,P, S 2 Cmax U.
()))
(1I)
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For the second part of the proof, consider the solution of (SOC3), namely (U3, Pj). optimality condition of this problem one obtains the following inequality:
Using the
P3 20) V2T2
Cmax U
Eur,
KL(((-
u) P3 (-12, w))
V
(12)
SLUUi 22T2
Cmax
U
Eu,
KL u) u) u))
Using the results in (11) and (12), one can then show the following chain of inequalities:
L.U,,P.c, 20U c,zo) -2T UmaxU
V Buzu
KL(P(-12z, u) Pj (-12, u))
ro) c =LUU c,
V2T2. Cmax V Bun
KL(P(-Zz, (1 w))
-2V2T Cmaxx U. V Bezu KL(P(H12,, u)ll
(13)
2LUU3, Pe. c, xo) + -2T2 CmaxU BBnu
(L(P(-1a, w) (+1x, w))
-2V2T22 Cmax U. V KL(P(-10, u)ll u) (-1z, w))
2LUU3, P,c, - 2V2212 Cmax
E KL(P(-1a, u)ll (+12, w))
where Uf is the optimizer of (SOC1) and (U3, P3) is the optimizer of (SOC3).
Therefore by letting \3 = /2T²2 CmaxU U and Rg RP(P) = Esy, KL(P(-lar, ()|/ w)) and by combining all of the above arguments, the proof of the above lemma is completed.
A.2 PROOF OF LEMMA 2
For the first part of the proof, at any time-step 2 1, for any arbitrary control action sequence - and any model P. consider the following decomposition of the expected cost:
t-1
B(c(),,u)) P,ro] = J0n-1EE
II
d (T Me-1)
k=1
f 2EE =4-E2
dD dD(ss|2))a((
ZEE
dG(aa 1. 1)
Now consider the following cost function: B[e(#)-1, 4-11) + c(Ee, ui) P, xol for 1 > 2. Using the above arguments, one can express this cost as
Eic(21-1 11 + clas, u.) P. moo
30. 28X(-2 II
(F(21-2/0022) 11112 182
dF 11162-11//22100 22-2)
22_002
(((
10 u ))((e, ui)
1-2
II dP km1
aB(2)210122)
2EXT-1
2EZ
J dF ( ((E--1) / 21E2
S -
II dP(zs/oo-1 DIv (f
1-2
(E(2)=r12)F((11 ue-2) 2EZ
Ciax
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By continuing the above expansion, one can show that E L(U,P,c,
E [L
ST
Cmax
Div((E (- us Il( (- ut)) 2.
2V2T Cmax
E
KL((E u))((F ()(+,, ue)) P,
T
SV2T2 Craax
KL((E ()) (F E) u.)) P,
T
1=0
where the last inequality based on Jensen's inequality of / function.
For the second part of the proof, following similar arguments as the second part of the proof of Lemma 1, one can show the following chain of inequalities for solution of (SOC3) and (SOC2):
L(U3. Ps, c0)
F3, 2 B [T(U3, c,
V2T2
Cmax U.
Bz,
KI(((F (1) (3)(,,,)))
-B FL, 20)
vol + v2T²..
Cmax U
KL((E} P)) E3) u))
-2V2T2 Cmax U. V Faz. KI,((Fig
(-)
E3)(,lz,)))
FE Ež, + v²T²
CmaXU V Ben
KL((E;
o E3)(,, u))
-2V2T2 Cmax U. V VEn KL((F' o (5) u))(( (3)(kk, u))
P2, - -22 22 Cmax U.
Ezza
KL((E)
(- u)l(( (3) u))
N2
(g()
(14)
where the first and third inequalities based on the first part of this Lemma, and the second inequality is based on the optimality condition of problem (SOC2). This completes the proof.
A.3 PROOF OF COROLLARY
11) To start with, the total-variation distance DTV (Jwex d' ()) (11,))) can be bounded by the following inequality using triangle inequality:
( W'E
DTV
( dP dP(z' E)(
DrV ( d'(a'll ())((==)()))) /
(,,
<DIv
W'EE
<DTV dP(c'r,,) E)(z,,))) DTv
N'EX where the second inequality follows from the convexity property of the Drv-norm (w.r.t. convex weights E(-lx'), Vz'). Then by Pinsker's inequality, obtains the following inequality:
( E( E)(,,,,
DIv
(15)
$1 2KL (
dP(r'|z, () (PP ()) + /2LL (P(-1z, u)ll w)))p(-12,.))
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We now analyze the batch consistency regularizer:
(2((P) E.n, [KL(E())))II(P (-(,,,
and connect it with the inequality in (15). Using Jensen's inequality of convex function x log for any observation-actin action pair (z, u) sampled from U, one can show that
J dP(c'|z, (z, u)
/ [E(2'(z') log ( dP(a'la l (P(alle,)) w)
2EEZ
(16)
{ dP(c'|z,
/ dE(''|z') (E(z'lz'))
Fee
22EZ
Therefore, for any observation-control pair (æ, u) the following inequality holds:
(L
KL
Vxex dP(x') ()) (1r,))))
= =I dP(x'le,,))
/ dE(='(z') log ( dP(clle,
JUE
ZEZZ
-f dP(x'l u) log (g(2' u))
(17)
WEX
$ $ JurE dP(r'(z,u) / dE(('((')) log (E(z12²)) Jre
dP(<'la u) log (g(2'|<, u))
ZEZZ (F(B( ()(,),
By taking expectation over (x,, u) one can show that
KL f
dP(x'Ie, ()E((1=))((F E)(lk,, (X)
Exgu
the lower bound the batch consistency regularizer. Therefore, the above arguments imply that
Drv (L dP(z'la E)("(( E)( & (2 RT + RP(P).
(18)
The inequality is based the property that /a + vb v2Va+b.
Equipped with the above additional results, the rest of the proof on the performance bound follows directly from the results from Lemma 2, which here we further upper-bound Drv (JeEx (a E))( E)((l,,,)) w)). when P
A.4 PROOF OF LEMMA 3
For the first part of the proof, any time-step > 1, for any arbitrary control action sequence T=) and for any model P. consider the following decomposition of the expected cost:
Elc(2,, u) P, Cmax
II (4 ((1101-1,
A=1
+
aP(xa/0k-11 4x-1) II k=
dE(2) 11 u--1)
dD (0cal21) (00e) uc)
EX!-IEX
ZEZ 22_EE
FEE
dG(z, 06-1,9
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Now consider the following cost function: Blc(Ee-1, u-11 + c(s,, ui) P, for t 2. Using the above arguments, one can express this cost as
E, 1) + daa, mc) P. P.
II
2
dP(ca/
dE(ci-2 22-2)
/ dF( uc-2)
10.1-2xx--
21-1
Am1
(Z1-1,
dD(a)-1/z-1) /
dE
u ) 21)
Cmax
Us
2011-3 k=1
1-2 dP(malaa-
dF
2EX 20:1-220 20:1-2Exi-1
22222
k=1
dF dF(z-1122-2, 1()-)1
11-1
2222
Cmax
5=l
3 kml
DAv (s (Plal/me-2, (12 / aE2
+ Cmax
d (x,,
2( ux-2))
Continuing the above expansion, one can show that [B [BU, 20) E, ((U,P.
(P uz) un) un)) + Divi f (n) w))) I P,
<1 Cmax E
T
J'E
21)) + / SwEx
Sv2T2 Canxx
E)(2,,, uc)) P, zu
T
<V2T2 Cmax
(KKK F)((,,, uz)) P,
<2T² -Inax
I2
(L) u.)
( ( P,
where the last inequality is based on the fact that vā vb v2va + b and is based on Jensen's inequality of v() function.
For the second part of the proof, following similar arguments from Lemma 2, one can show the following inequality for the solution of (SOC3) and (SOC2):
L(Ui, 2o) [R F2, 20)
Eg /2T². Caax U. V (P:)
=E[L(UG 2, 20)
Eg, + /2T² Cmax U. PP) + 33g
22/2T22 CmaxU
V 12 (PT) + (Px)
E2 2 -max U. v (P + (P3 (19)
> LE Fr c, 20)
-2/212 C U. V 22 P2)) 3R
LLUU; P,c, - 22T22 Cmax U. v 2Rg (?7) + (P))
Az
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where the first and third inequalities are based on the first part of this Lemma, and the second inequality is based on the optimality condition of problem (SOC2). This completes the proof.
A.5 PROOF OF LEMMA 4
Recap of the Result: Let (Ui.c. P..c) be a LLC solution to (SOC-LLC) and UJ a so- lution to (SOC1). Suppose the nominal latent state-action pair {(z,, 4)){ satisfies the con- dition: (zz; ue) ~ ((222 (2, 8)1), where {[22,, 22 1T-1 is the optimal trajectory of prob- lem (SOC2). Then with probability 1 - 7, we have L(U P,, 2 L(Uüic P, P,,,
2uucy Ro (a(PPc)) Pucc(PiL).
Discussions of the effect of ō on LLC Performance: The result of this lemma shows that when the nominal state and actions are ò-close to the optimal trajectory of (SOC2), i.e., at each time step (z,, uz) is a sample from the Gaussian distribution centered at (22. U2,2) with standard deviation 6, then one can obtain a performance bound of LLC algorithm that is in terms of the regularization loss RuiC- To quantify the above condition, one can use Mahalanobis distance (De Maesschalck et al., 2000) to measure the distance of (z,, u,) to distribution (({ u²,2), 82 I), i.e., we want to check for the condition:
[2 uz) - -(2,, 22 <e, Vt,
for any arbitrary error tolerance e' > O. While we cannot verify the condition without knowing the optimal trajectory { u2, u the above condition still offers some insights choosing the
parameter ó based on the trade-off of designing nominal trajectory {(z,, u)) and optimizing RLLC. When 8 is large, the low-curvature regularization imposed by the RLLC regularizer will cover large portion of the state-action space. the extreme case when 8 -> oc, RyEC can be viewed as a regularizer that enforces global linearity. Here the trade-off is that the loss RLIC is generally higher, which in turn degrades the performance bound of the I.I.C control algorithm in Lemma 4. On the other hand, when d is small the low-curvature regularization in RLLC only covers a smaller region of the latent state-action space, and thus the loss associated with this term is generally lower (which provides a tighter performance bound in Lemma 4). However the performance result will only hold when (ze; u,) happens to be close to (z2,e> u²,,) each time-step E (o....,T
Proof: For simplicity, we will focus on analyzing the noiseless case when the dynamics is deter- ministic (i.e., Dn = 0). Extending the following analysis for the case of non-deterministic dynamics should be straight-forward.
First, consider any arbitrary latent state-action pair (z, u), such that the corresponding nominal state-action pair (z, u)) is constructed by z oz, u = u 8u, where (dz, 8u) is sampled from the Gaussian distribution N/ (0, 82 I). (The random vectors are denoted as (8z', ,d')) By the two-tailed Bernstein's 's inequality (Murphy, 2012), for any arbitrarily given E (0, 1] one has the following inequality with probability ny.
(z, u) + A(z, u)d= [
V 2 log(2/1) (6x' /1 A(z, u)22' + f2
+ + (a0e:,aay)ww(0o.o////(.... A(2,
B(z, u)on' fz(2,
1/2
<(1 + 222lgl
E (1lfz A(z, + B(z,u) u1)]
Ruc (1
The second inequality is due the basic fact that variance is less than second-order moment of random variable. On the other hand, at each time step f {0, ........ ...,, T - 1} by the Lipschitz property of the immediate cost, the value function V2(2) min E (TT(2) + Σ C (27, u,) 2 = 2 is also Lipchitz with constant (T f 1) Using the Lipschitz property of V4+1, for any (2,u)
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and (óz, du), such that (z,u) = (z 82, 6u), one has the following property:
|Vi+1 A(z, u)8z B(z, u)bu) (f2 u))l <T - () (z u) + A(z, 2) + uz)u fz(2,
(20)
Therefore, at any arbitrary state-action pair (2, ū) for sample (δz, ōu) ~ (O(0, 82 T), the following inequality on the value function holds w.p.
oz, and du with Gaussian
(i+1(f2(2, 0)) Vi+1 (z' A(z, ((z,u)ōu) (T -t) (1 + /2 V RuLC
which further implies
(2 0) + (2z)
(2 0) + Vi41(2' + A(z,u)82 + B(z,u))u) (T ttoppp + v 2 2ogg(2/))
BLLC (Pl2, i).
Now let ū' be the optimal control w.r.t. Bellman operator Z, ( any latent state ž. Based on the assumption of this lemma, at each state the nominal latent state-action pair (z, u) is generated by perturbing (2, ü²*) with Gaussian sample (5z, 5u) (N0, 5² that is in form of 3 82, 8u. Then by the above arguments the following chain of inequalities holds w.p. 1 ry:
T. (V++ == mina (3, ú) + (e+1(f2(2,
=c((2, it) + (1+1
20((2, ū²) + Vi+1
A(z, u)dz B(z, u)ōu)
IV+1 + A(z, u) u)) V++ (Jz(E,,)) Alz, u u)dz + B(z, u)uu
20(8,, u + ōu) + V1+
(21)
- (T - 1)- + v 2log(2/11)
Ruc P|zz, u)
max 2,a
2 min (3, 5u) + V2+1(2Z(22 + u)z, + B(z, u u)du)
6u
- (T - ()-ppp + / /2log(2/1))
max RI. (p1z,u) 2,2
Recall the LLC loss function is given by
PLc((P) Em,. E [BLLC(PIz, u) |2
E
Also consider the Bellman operator w.r.t. latent SOC: T: [V (z)
min, (2, u) V(22(2, u)). and
the Bellman operator w.r.t. ILC.: (e, minga, 7. (2, du + +u) + (Jz(z, + A(z, u)dz B(, ou). Utilizing these definitions, the inequality in (21) can be further expressed as
T[V++11(3 TTiLc (V4rl(()))))))---- (T' Cmux (1 + v logg() UX RLcC(P),
(22)
This inequality due to the fact that all latent states are generated by the encoding observations, i.c. 2 ~ E.((r), and thus by following analogous arguments as in the proof of Lemma 1, one has
max PLIC < UUEEn
E RILC(
E U (LLc())))
Therefore, based on the dynamic programming result that bounds the difference of value function w.r.t. different Bellman operators in finite-horizon problems (for example see Theorem 1.3 in Bertsekas (1995)), the above inequality implies the following bound in the value function, w.p. r):
min L(U, F,C, U.P
T-1
2IUULc Dh c,
Σ 1=1
(T - ChipCmax T -T + 22log(2T/n) VUX. v Rucc(Pice
2 LUULC Prx Picc C, 20) - -T Cmpx (1 22log(2T/n)) VUX Rucl(Pic)
(23)
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Notice that here replace in the result in (22) with n/T. In order to prove (23), utilize (22) for each E {O..... T - 1,, and this replacement is the result of applying the Union Probability bound (Murphy, 2012) (to ensure (23) holds with probability 17).
Therefore the proof is completed by combining the above result with that in Lemma 3.
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B THE LATENT SPACE ILQR ALGORITHM
B. PLANNING IN THE LATENT SPACE (HIGII-LEELL DESCRIPTION)
We follow the same control scheme as in Banijamali et al. (2018). Namely, we use the iLQR (Li & Todorov, 2004) solver to plan in the latent space. Given a start observation Tstart and goal observation Fooal corresponding to underlying states (sttart Sgoal and Zgoal. Then, the procedure goes as follows: we initialize a random trajectory (sequence of actions), feed it to the iLQR solver and apply the first action from the trajectory the solver outputs. We observe the next observation returned from the system (closed-loop control), and feed the updated trajectory to the iLQQ solver. This procedure continues until the it reaches the end the problem horizon. We use a receding window approach, where at every planning step the solver only optimizes for fixed length of actions sequence, independent of the problem horizon.
encode the observations to retrieve Astart
B.2 DETAILS ABOUT ILQR IN THE LATENT SPACE
Consider the latent state SOC problem
T-1 C0( un)
min E Cr (TT))) + Σ
20
At each time instance E (0,...,, 7) the value function of this problem is given by
T-1
(z)))=or(a) Vi(z) min E (T()))
(.. 22
Vt <T
(24)
Uer
Recall that the nonlinear latent space dynamics model is given by:
2++1 F F(2,, ue, we) == (z,, u4) + Fg Wt, We NO0 I), Vt V200,
(25)
where F2, uz) is the deterministic dynamics model and F.T F. is the covariance of the latent dynamics system noise. Notice that the deterministic dynamics model Fu(2t, u4) is smooth, and therefore the following Jacobian terms are well-posed:
oF, (F, B(z,u) Oz
OF, (F Vz R"= Vu ER..
A(z, u)
Ou
By the Bellman's principle of optimality, at each time instance E (O...., 1} the value function is a solution of the recursive fixed point equation
Vi(2) min Qı(z,u),
(26)
where the state-action value function at time-instance w.r.t. state-action pair (2,, uz) (2, u) is given by Q2(2,u) u) = (z u) + Ew, (F+1(F(Ze, ue, Wx)) 2e z, Ue = u].
In the setting of the iLQR algorithm, assume we have access a trajectory of latent states and actions that is in form of {2,, u,, Ze+ 1 At each iteration, the iLQR algorithm has the following steps:
1. Given a nominal trajectory, find an optimal policy w.r.. the perturbed latent states
2. Generate sequence of optimal perturbed actions that locally improves the cumulative cost of the given trajectory
3. Apply the above sequence of actions to the environment and update the nominal trajectory
4. Repeat the above steps with new nominal trajectory
Denote by 82, 21 2, and 5u, U4 ut the deviations of state and control action at time step t respectively. Assuming that the nominal next state 21+1 generated by the deterministic transition Fu (Zz, u₂) at the nominal state and action pair (z,, uz), the first-order Taylor series approximation of the latent space transition is given by
824+1 21+1 Z+41 A(zi, un )0zz B(2,, u. ou, + Fg + ((), ou.)li), wi ~ (00 T). (27)
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To find a locally optimal control action sequence ut (ó2z) ue Vt, that improves the cumulative cost of the trajectory, we compute the locally optimal perturbed policy (policy w.r.t. perturbed latent state) (624)) TT1 that minimizes the following second-order Taylor series approximation of Q. around nominal state-action pair (Z1, u+), V1 E To.....T-1]:
F. Fo Qualzr, up) Qi,a(ze, ue
Qr U1) Qrsa (Ze, u.) Qi,uz (2,, Ue)
Qrn uz) Qr,us (2e, uc) Qi 24
Qe(z,, ue) (z(,
dzt dūt
δzg (28) 8uee
where the first and second order derivatives of the Q o--fucction are given by
Öe,(Ze, ui)
(1, ux)
+ A(2,, u.) (il))Ve+ette Ve+1, u.)
Ōz
[7, u.)
Qz, u+)
+ B(2,, ue) V1+1, ui)
Du
Ceee ui)
Q1,z= we)
+ A(2z, V u4) A((,, Us)
0²2 (21, U1)
Qt,u(21, we)
+ B(zi, u.) V1+1,22 Ue A(()) Ut)
Oudz
02 czle uI) Ou²
Qt,un(ze, un)
+ B(z,, un) VV 1141 (zz, un (z,, ue
and the first and second order derivatives of the value functions are given by
TVV++1
(F(z,, uz, w)) V1+1,==(Z1, u.) = E
0 (F(zi, un w.) Oz2
Ve+1,= = Eu
0=
Notice that the Q-function approximation Qt in (28) is quadratic and the matrix (21, ue)) Qt, (Ze, Ut) Qu.uu (2,, uz)
Quana (Ze, u.)
is positive semi-definite Therefore the optimal perturbed policy
Thet has the following closed-form solution:
(-), arg du:
arg min Qe(ze, = (62z) ke( + Ke(ze, (1 2,00
(29)
where the controller weights are given by (2 (b) = uQ (21, u.)) Qt, u.) and Ke(ze, uz) = (Qt,um (2 uz)) Qt (Ze, u.). Furthermore, by putting the optimal solution into the Taylor expansion of the Q-function Qo we get
1
Qin (2,, u.) Qi (2,, uz)
(Qi u.))
'1
Qe(ze, u) Qe u,) 2 622
Qi,z2 we)
822
where the closed-loop first and second order approximations of the Q-function are given by (z,, u.) C. Qt, u.) "aauu.TQuuuua "Q,auuuz (4)) 1QQ,, (2t,
Qi (ze, u.) = Qz, u.) ke uz) Qu (ze, us Ke (z u:), (z,, u.) (2,, u.) ) K. u.) T (Zt, u. Ki(z,, uz). Notice that at time step t the optimal value function also has the following form: (2(2 = min Qz u) =60;(2(, ue 82,, (62.)) + 2e(ze,u))
1 [ Q (zi, un) Qi un)
(Qi (Ze, (Z22 u:)) (z.e u.)
8zz + Qi(zz,, u.).
2 ózz
(30)
Therefore, the first and second order differential value functions can be V1, (zz u.) 2i (Ze, u.), (z,zz(ze, u:) = Qi,2a u:), and the value improvement at the nominal state at time step / is given by Qi. (z., un) + Qe(2,, u.).
Vi(2,) = 2
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B.3 INCORPORATING RECEDING- HORDIIN TO ILQR
While iLQR provides an effective way of computing sequence of (locally) optimal actions, has two limitations. First, unlike RL in which an optimal Markov policy is computed, this algorithm only finds sequence of open-loop optimal control actions under given initial observation. Second, the iLQR algorithm requires the knowledge of nominal (latent state and action) trajectory every iteration, which restricts its application to cases only when real-time interactions with environment are possible. In order to extend the iLQR paradigm into the closed-loop RL setting, we utilize the concept of model predictive control (MPC) (Rawlings & Mayne, 2009; Borrelli et al., 2017) and propose the following iLQR-MPC procedure, Initially, given an initial latent state 2o we generate a single nominal trajectory: {(z,, u, Z++1)) ) whose sequence of actions is randomly sampled, and the latent states are updated by forward propagation of latent state dynamics (instead of interacting with environment) i.e., 20 20, Ze+1 [z,,, Uc Wt), Vt. Then at each time-step k 2 O, starting
at latent state 2ę we compute the optimal perturbed policy { (:) T-1 using the iLQR algorithm
with T k lookahead steps. Having access to the perturbed latent state özę 22 2k. we only deploy the first action " = , (6zx) + Ug in the environment and observe the next latent state
2k+1- Then, using the subsequent optimal perturbed policy (-)) we generate both the estimated latent state sequence { by forward propagation with initial state ōk+1 -k+11 and action sequence {u²
{u:)/== ) where u, = T2,,, (62)) + uc and 82, 2g 2g. Then one updates the subsequent nominal trajectory as follows: {(2,, Uc, 21+1 repeats the above procedure.
{(2,, n S4t1) and
Consider the finite-horizon MDP problem min,, E (TT) + E ( uz) Te, P. xo
where the optimizer is over the class of Markovian policies. (Notice this problem is the closed-loop version of (SOC1).) Using the above iLQQR-MPC procedure, at each time step & one can construct Markov policy that is in form of
T
iLQR Te.iLQR- (PC.R == ui s.t. {ut
ILQR
FLOR 4T iLQR(L(U, P" P",,,,,)) 2)), with 2 E((xx)),
where ELQR(ccomme (U: P-,,)) denotes the iLQR algorithm with initial latent state 2. To understand the performance of this policy w.r.t the MDP problem, we refer to the sub-optimality bound of iLQR (w.r.t. open-loop control problem in (SOCI)) in Section 3.3, as well as that for MPC policy, whose details can be found in Borrelli al. (2017).
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Ut
2
1
Ft
F+++
Zt
T++1
Figure 3: PCC graphical model. Left: generative links, right: recognition links.
C
TECHNICAL PROOFS OF SECTION 4
C.I DERIVATION OF R NLLE BOUND (P
(P.Q) DECOMPOSITION
We derive the bound for the conditional log- log-likelihood log P(04t1/0,,u) uc).
log P ue) log /
P(1e41, 24, 24 (24, Ue d21
20,2e11
P 24, 20+1 ( Us (Er, Zet1 1e Fe+ Up
log BQ(snne4s1l
log Pla 22, 24+ 1 220, Uc
(zze Re+1 [ , We
((x (rr, P(ee+ 3e+ I2e, Ue ((
(b) Q(141 (a1))
log Q ( 14441 1Q &441, us)
Qz (a.
Eq (m++1)
[oog P( (2+21
-EQeenlee 1)
DKI. (Q(z)) (0(2:lé441,00,, ue ((
+ H (
log P(2)41)z),))) un)
+ +E Q(2e+1 wets) Q(2e Se ,.,,))))
R Bound (P.Q)
Where (a) holds from the log function concavity, (b) holds by the factorization Qz z [ I1+1, uz) =
ae ue), and (c) holds by simple decompo-
sition the different components.
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C.2 DERIVATION OF R BounnD (P. Q) DECOMPOSITION
We derive the bound for the consistency loss &Consistency (P).
DKL DKL (((441|00-1))) /
PD(P)
P(eet1l24, We) P(ze
@ H (Q(8411(2441) (Q log S P(1141l2e, us P(zelza)des
P(e4t1l 22, ue P((
(Q(241)(1)1)) EQ(e441
log Ee(ze (1,1)
Q(2a 22+1, T2, u4)
(6)
log P(1 2,, uo) P(t,, xx))
H (Q(2(+1
Q(21 (2212241,,20 22, 21)
(
E((ennleen) DKL (Q(2(l241,,
24, (())((((
H (2(4+1 E ((an))))) log P( E1, me)
R2 Bound (P,Q)
Where (a) holds by the assumption that Q( | 2++1) function concavity, and (c) holds by a simple decomposition to the different components.
P(241 xt+1). (b) holds from the log
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D EXPERIMENTAL DETAILS
In the following sections we will provide the description of the data collection process, domains, and implementation details used in the experiments.
D.1 DATA COLLECTION PROCESS
(2,, To generate training and test sets, each consists of triples (2), Me, 2t+1), we: (1) sample an
underlying state st and generate its corresponding observation 2r, (2) sample an action ut, and (3) obtain the next state $t+1 according to the state transition dynamics, add it a zero-mean Gaussian noise with variance o² In, and generate it's corresponding observation Iz41.To To ensure that the observation-action data is uniformly distributed (see Section 3), we sample the state-action pair (st, We) uniformly from the state-action space. To understand the robustness of each model, we consider both deterministic (a 0) and stochastic scenarios. In the stochastic case, we add noise to the system with different values of and evaluate the models' performance under various degree of noise.
D.2 DESCRIPTION OF THE DOMAINS
Planar System In this task the main goal is to navigate an agent in surrounded area on 2D plane (Breivik & Fossen, 2005), whose goal is to navigate from a corner to the opposite one, while avoiding the six obstacles in this area. The system is observed through a set of 40 x 40 pixel images taken from the top view, which specifies the agent's ' location in the area. Actions are two-dimensional and specify the x y direction of the agent's 's movement, and given these actions the next position of the agent is generated by a deterministic underlying (unobservable) state evolution function. Start State: one of three corners (excluding bottom-right Goal State: bottom- right corner. Agent's 's Objective: agent is within Euclidean distance of from the goal state.
Inverted Pendulum - SpingUp & Balance This is the classic problem of controlling an inverted pendulum (Furuta et al., 1991) from 48 x 48 pixel images. The goal of this task is to swing up an under-actuated pendulum from the downward resting position (pendulum hanging down) the top position and to balance it. The underlying state st of the system has two dimensions: angle and angular velocity, which is unobservable. The control (action) is 1-dimensional, which is the torque applied to the joint of the pendulum. To keep the Markovian property in the observation (image) space, similar to the setting E2C and RCE, each observation 2s contains two images generated trne-frames (from current time and previous time). This is because each image only
from consecutive shows the position of the pendulum and does not contain any information about the velocity. Start State; Pole is resting down (SwingUp), or randomly sampled in ///6 (Balance). Agent's 's Objective: pole's angle is within / from an upright position.
CartPole This is the visual version of the classic task of controlling cart-pole system (Geva & Sitte, 1993). The goal in this task is to balance a pole on a moving cart, while the cart avoids hitting the left and right boundaries. The control (action) is 1-dimensional, which is the force applied to the cart. The underlying state of the system 8, is 4- dimensional, which indicates the angle and angular velocity of the pole, as well as the position and velocity of the cart. Similar to the inverted pendulum, in order to maintain the Markovian property the observation Jy is a stack of two 80 x 80 pixel images generated from consecutive time-frames. Start State: Pole is randomly sampled in + /6. Agent's 's Objective: pole's angle is within =7/10 from an upright position.
3-link Manipulator - Swing & Balance The goal in this task is to move a 3-link manipulator from the initial position (which is the downward resting position) to a final position (which is the top position) and balance it. In the 1-link case, this experiment reduced to inverted pendulum. In the 2-link case the setup is similar to that of arcobot (Spong, 1995), except that we have torques applied to all intermediate joints, and in the 3-ik case the setup is similar to that of the 3-lk planar robot arm domain that was used in the E2C paper, except that the robotic arms are modeled by simple rectangular rods (instead of real images of robot arms), and our task success criterion requires both swing-up (manipulate
final position) and balance." 12 The underlying (unobservable) state 8, of the system is 2N -imennioonl which indicates the relative angle and angular velocity at each link, and the actions are N -ddienniionll, representing the force applied to each joint of the arm. The
12 Unforttnately due copyright issues, we cannot test our algorithms on the original 3-link planar robot arm domain.
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state evolution is modeled by the standard Euler-Lagrang. equations (Spong. 1995; Lai et al., 2015). Similar the inverted pendulum and cartpole, in order maintain the Markovian property, the observation state Fy is a stack of two 80 x 80 pixel images of the N-link manipulator generated from consecutive fime-esmms. In the experiments we will evaluate the models based on the case of N (2-link manipulator) and N = 3 (3-link manipulator). Start State: 1st pole with angle τ, 2nd pole with angle 2 2 and 3rd pole with angle t/3, where angle æ is resting position. Agent's 's Objective: the sum of all poles' angles is within + from an upright position.
TORCS Simulaotr This task takes place in the TORCS simulator (Wymann al., 2000) (specifi- cally michegan race track, only straight lane). The goal of this task is to control car so it would remain in the middle of the lane. We restricted the task to only consider steering actions (left right in the range of [-1, 1]). and applied a simple procedure to ensure the velocity of the car is always around 10. We pre-processed the observations given by the simulator (240 320 RGB images) to receive 80 80 binary images (white pixels represent the road). In order to maintain the Markovian property, the observation state 22 is stack of two 80 x 80 images (where the two images are 7 frames apart- chosen
that consecutive observation would be somewhat different). The task goes as follows: the car is forced steer strongly left (action=1) l), or strongly right (action=-1) for the initial 20 steps of the simulation (direction chosen raandolyy. which causes it to drift away from the center of the lane. Then, for the remaining horizon of the task, the car needs to recover from the drift, return the middle of the lane, and stay there. Start State: 20 steps of drifting from the middle of the lane by stecring strongly left, or right (chosen randomly). Agent's 's Objective: agent (car) is within Euclidean distance of 1 from the middle of the lane (full width of the lane is about 18).
D.3 IMPLEMENTATION
In the following we describe architectures and hyper-parameters that were used for training the different algorithms.
D.3.1
TRAINING PAREMEERRRE AND REGULIZERS
All the algorithms were trained using:
Batch size of 128.
ADAM (Goodfellow et al., 2016) with a 5- 1111100-000 = 0.9, 32 = 0.999, and E = 10 10-8
L2 regularization with coefficient of 10-3
Additional VAE (Kingma & Welling, 2013) loss term given by AAE
-E,(= [log p(z]z) + DxL (g(2)=))lp(2)). where p(2) (O,, The term was added with very small coefficient of 0.01. We found this term to be important to stabilize the training process, there is explicit term that governs the scale of the latent space.
E2C training specifics:
X from the loss term of E2C was tuned using parameter sweep in (0.25, 0.5, 1}, and was chosen to 0.25 across all domains, as performed the best independently for each domain.
PCC training specifics:
Ap was set to across all domains.
Àe was set to be across all domains, after it was tuned using parameter sweep in
(1. 10) on the Planar system.
Aeur was set to be 1 across all domains without performing any tuning.
u). for the curvature loss, were generated from (2, u) by adding Gaussian noise
(00, 0.13), where
0.1 was set across all domains without performing any tuning.
Motivated by Hafner et al. (2018), we added a deterministic loss term in the form of cross entropy between the output of the generative path given the current observation and action (while taking the means of the encoder output and the dynamics model output) and the observation of the next state. This loss term was added with a coefficient of 0.3 across all domains after it was tuned using parameter sweep over {0.1, 0.3, 0.5) on the Planar system.
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D.3.2 NETWORK ARCHITECTURES
We next present the specific architecture choices for each domain, For fair comparison, The numbers of layers and neurons of each component were shared across all algorithms. ReLU non-linearities were used between each two layers.
Encoder: composed of a backbone (either a MLP or a CNN, depending on the domain) and an additional fully-connected layer that outputs mean variance vectors that induce diagonal Gaussian distribution.
composed of a backbone (either a MLP or a CNN, depending on the domain) and an additional fully-connected layer that outputs logits that induce a Bernoulli distribution.
Decoder:
Dynamical model: the path that leads from {2,, We } 21+1- Composed of MLP backbone and an additional fully-connected layer that outputs mean and variance vectors that induce diagonal Gaussian distribution. We further added a skip connection from 2 and summed it with the output of the mean vector. When using the amortized version, there are two additional outputs A and B.
Backwards dynamical model: the path that leads from ( Wr, vz Zt. each of the inputs goes through fully-connected layer with {N,,, N₂ neurons, respectively, The outputs are then concatenated and pass though another fully-connected layer with Njoim neurons, and finally with an additional cunlecco layer that outputs the mean and variance vectors that induce diagonal Gaussian distribution.
Planar system
Input: 40 40 images. 5000 training samples of the form (oee, My Tr+11
Actions space: 2-dimensional
Latent space: 2-dimensional
Encoder: Layers: 300 units - 300 units 4 units (2 for mean and for variance)
Decoder: Layers: 300 units 300 units 1600 units (logits)
Dynamics: 3 Layers: 20 units - 20 units 4 units
Backwards dynamics: N₂ 5, .N. = 5, N₂ = 100 - -Njonnt 100 44 units
Number of control actions: the planning horizon T 40
Inverted Pendulum -wing Up Balance
Input: Two 48 x 48 images. 20000 training samples of the form (Iz, uy T++1)
Actions space: 1-dimensional
Latent space: 3-dimensional
Encoder: Layers: 500 units 500 units 6 units (3 for mean and for variance)
Decoder: Layers: 500 units 500 units 4608 units (logits)
Dynamics: Layers: 30 units 30 units units
Backwards dynamics: N, 10, N, 10, N₂ 200 Nyaim 200 6 units
Number of control actions: or the planning horizon T 400
Cart-pole Balancing
Input: Two 80 80 images. 15000 training samples the form (Ie, uy Tc+1)
Actions space: 1-dimensional
Latent space: 8-dimensional
Encoder: 6 Layers: Convolutional layer: 32 x 5: stride (1 Convolutional layer:
32 x 5 x 5; stride (2, 2) -Convllutional layer: 32 x 5 x 5; stride (2, 2) Convolutional
layer: 10 x 5 x 5; stride (2,2) 200 units 16 units (8 for mean and 8 for variance)
Decoder: Layers: 200 units 1000 units 100 units Convolutional layer: 32
stride (1, Upsampling (2,2) convolutional layer: 32 x x 5; stride (1, 1) Upsampling
(2,2)- Convolutional layer: 32 5 x 5; stride (1, 1) -00--UUsan Upsampling (2, 2) Convolutional
layer: 2 x 5 5; stride
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Dynamies: Layers: 40 units 40 units 16 units
Backwards dynamics: N₂ 10, N, 10, N₂ 300 Noint 300 16 units
Number of control actions: the planning horizon T = 200
3-link Manipulator - Swing Up & Balance
Input: Two 80 80 images. 30000 training samples the form (Ie ue T++1)
Actions space: 3-dimensional
Latent space: 8-dimensional
Encoder: Layers: Convolutional layer: 62 x x 5; stride (1, 1) Convolutional layer:
32 x 5 X 5; stride (2, 2) Convolutional layer: 32 x 5 x 5; stride (2. 2) Convolutional layer: 10 5 X 5; stride (2, 2) 500 units 16 units (8 for mean and 8 for variance)
Decoder: 6 Layers: 500 units 2560 units 100 units Convolutional layer: 32 5 x 5; stride 1) Upsampling (2, 2) convolutional layer: 32 x 5; stride (1, 1) Upsampling (2,2) Convolutional layer: 32 x 5 x 5; stride (1, 1) Upsampling (2, 2) Convolutional
layer: 2 x 5 5; stride (1,
Dynamics: 3 Layers: 40 units 40 units 16 units
Backwards dynamics: N₂ 10, N,, 10, N, = 400 Nyain 400 16 units
Number of control actions: or the planning horizon T 400
TORCS
Input: Two 80 80 images. 30000 training samples of the form (ae, uy Ft+11
Actions space:
1-dimensional
Latent space: 8-dimensional
Encoder: Layers: Convolutional layer: 32
stride (1, 1) Convolutional layer:
32 x 5 x 5; stride (2, 2) CConvolutioaal layer: 32 x 5 x 5; stride (2, 2) Convolutional layer: 10 5 x 5; stride (2, 2) 200 units 16 units (8 for mean and 8 for variance)
Decoder: 6 Layers: 200 units 1000 units 100 units Convolutional layer: 32 5 x 5; stride 1) -ppsmpling (2, 2) convolutional layer: 32 x 5: stride (1, 1) Upsampling (2, Convolutional layer: 32 x 5; stride (1, 1) Upsampling (2, 2) Convolutional
layer: 2 x 5 5; stride (1, 1)
Dynamics: Layers: 40 units 40 units 16 units
Backwards dynamics: N; 10, N, 10, N, = 300 - N 300 16 units
Number of control actions: or the planning horizon T' 200
E ADDITIONAL RESULTS
E.
PERFORMANCE ON NOISY DYNAMICS
Table 3 shows results for the noisy cases.
Table 3: Percentage of steps in goal state. Averaged over all models (left), and over the best model (right). Subscript under the domain name is the variance of the noise that was added.
Domain
RCE (all)
(al)
PCC (all)
RCE (top
E2C (top 1)
PCC
Planarj
1.2 0.6
0.6 0.3
17.9 = 3.1
3.5 1.2
6.1 = 0.9
44.7 = 3.6
Planara
0.4 0.2
1.3 0.9
14.5 2.3
E.5
15.5 2.6
29.7 2.9
Pendulum,
6.4 0.3
23.8 = 1.2
16.4 = 0.8
8.1
36.1 1 0.3
29.3 = 0.2
Cartpole
8.1 0.6
6.6 0.4
9.8 = 0.7
20.3 1 11
16.3 1 0.4
17.9 0.8
3-link
0.3 0.1
020
0.5 0.1
1.3 0.2
0:0
1.8 0.3
27

---

Published as a conference paper at ICLR 2020
E.2 LATENT SPACE REPRESENTATION FOR THE PLANAR SYSTEM
The following figures depicts instances (randomly chosen from the 10 trained models) of the learned latent space representations for both the noiseless and the noisy planar system from PCC, RCE, and E2C models.
Figure 4: Noiseless Planar latent space representations using PCC.
Figure 5: Noisy (o² 1) Planar latent space representations using PCC.
Figure 6: Noiseless Planar latent space representations using RCE.
Figure 7: Noisy (a² 1) Planar latent space representations using RCE.
Figure 8: Noiseless Planar latent space representations using E2C.
Figure 9: Noisy (o² = 1) Planar latent space representations using E2C.
28


Reviews:
[{"review_id": "r1xykC03qS", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper considers learning low-dimensional representations from high-dimensional observations for control purposes. The authors extend the E2C framework by introducing the new PCC-Loss function. This new loss function aims to reflect the prediction in the observation space, the consistency between latent and observation dynamics, and the low curvature in the latent dynamics. The low curvature term is used to bias the latent dynamics towards models that can be better approximated as locally linear models. The authors provide theory (error bounds) to justify their proposed PCC-Loss function. Then variational PCC is developed to make the algorithm tractable. The proposed method is evaluated in 5 different simulated tasks and compared with the original E2C method and the RCE method. \nThe paper is well-written. \n\nPros:\n- The idea in this paper is quite original. The three principles used to formulate the loss function provide some new insights.\n\n- The authors have proposed a theory to justify the use of their loss function. The technical quality of this part seems solid.\n\n- Simulations have been used to show that the proposed PCC method outperforms E2C and RCE.\n\n-The paper is well written. \n\nCons:\n- The tasks in this paper are not that complicated. It is unclear whether the proposed method outperforms other model-based RL methods such as Solar and DSAE for practical robotic applications. More comparisons are needed.\n\n- It is also not that clear why one wants to SOC3 to be close to SOC1 in the first place. It seems the true optimization problem should be posed on the space of the original state s. SOC1 is just a surrogate problem for the original problem.\n\n- There seems to be a gap between the proposed theory  and the algorithm implementation. This makes the theory part less useful.\n\nOverall, I think the idea in this paper is interesting. The authors have made a serious effort in coming up principles for model-based RL control. But at this moment it is not that convincing the proposed method will be the best model-based RL method for practical robotic applications. If the authors can address my comments, I will be willing to increase my score.\n\n\n\nMinor Comments:\n\n- It seems that for the task the authors have tested their method, it is not that difficult to directly estimate the state. Am I correct here? Can the authors make a comment on this? How to compare their approach and a more direct control approach using estimation of state s? \n\n- I have never seen the curvature principle in any control papers. Any control reference on why this is a good principle? It seems that the linearization works well when the control inputs are around the reference points. Does the curvature really matter that much for ILQR to work? \n\n- How to justify the Markovian assumption on x? Just by observation or there is a more principle way to test this assumption on the buffered images?\n\n====================================================\nPost-Rebuttal:\nAfter reading the authors' response, I am changing my score to weak accept. Lemma 4 is nice. I have not seen anything similar to this in the controls literature before. The authors have addressed most of my concerns. I still have a few comments for preparing the final version of this paper. \n1. I still don't see why SOC1 is the \"original problem.\" Yes, it is assumed that the true state cannot be directly observed. But if the observations are Markov eventually, then some estimated version of the states can be obtained, right? I think treating SOC1 as the original problem is one possible way of doing things and clearly the authors have built a principled framework for doing things in this way. But treating SOC1 as the original problem seems not the only way of doing things. I hope the authors can clarify this and do not oversell the proposed approach. \n2. I think it is still worth comparing SOLAR and PCC empirically. This will help the readers to choose algorithms when they need. \n3. The comment on the verification of the Markov assumption is hand-waving.  The authors said \"A simple test would be to see if a control algorithm with the Markovian assumption works well with our representation or not.\" Does this mean that the users will not be able to verify this assumption before using the proposed approach to obtain controllers? It will be helpful if the authors can explain this step for one specific example in details.\n", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I made a quick assessment of this paper.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "HylFTIdRYB", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": null, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper considers from a high level the problem of learning a latent representation of high dimensional observations with underlying dynamics for control.  The authors specifically describe some desiredata for latent representations for LLC algorithms. The authors rigorously construct a learning framework that can satisfy the desiredata and then show how this can be tractably instantiated. \n\nThe paper overall is clear, however there is many equations in 4.2  with heavy subscritping making it sometimes difficult to read. The authours could attempt to better highlight the more critical parts of their propositins (e.g. eq. 8/9). \n\nThe methodology and insights appear novel and well motivated, however I am not familiar with many of the prior work.  The experiments compared to competing methods  show substantial improvement. The authors also motivate well why these improvements over the existing methods should occur and provide ablations to validate all the components of the final loss. Overall the paper appears very solid and may motivate insights and research  in more complex model based control and planning \n", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I did not assess the derivations or theory."}, "extra_text": {}}, {"review_id": "HkxhB8v8KS", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This work proposes a regularization strategy for learning optimal policy for a dynamic control problem in a latent low-dimensional domain. The work is based on LCE approach, but with in-depth analysis on how to choose/design the regularization for the \\hat{P} operator, which consists of an encoder, a decoder, and dynamics in the latent space. In particular, the author argued that three principles (prediction, consistency, and curvature) should be taken into consideration when designing the regularizer of the learning cost function - so that the learned latent domain can serve better for the purpose of optimizing the long-term cost in the ambient domain. \n\nThe paper is well written and pleasant to read. One possible shortcoming is that the notations are a bit dazzling. It is almost impossible to follow the notation when first reading this paper. The proofs are very lengthy and thus the reviewer did not check in detail. \n\nThe reviewer has several question:\n\n1) Of course SOC2 makes sense. But what if one models the whole problem as an HMM, and perform control algorithms in the hidden domain of the HMM (and the hidden states can be of much smaller alphabets compared to the observable states), will there be any fundamental difference? Of course learning an HMM is challenging, but approachable. Any comments?\n\n2) The three design principles make sense, but may need more elaboration. For example, it is a bit unclear why f_Z should be with low curvature -- does it mean that you wish the control problem in the latent domain is more like a linear dynamical system, so that the LLC algorithm works better? The argument is a bit unclear, since \"locally linear\" is not a rigorous term. Any smooth function is ``\"locally linear\". Here, how to measure the difficulty of the latent control problem may need more discussion.\n\nMinor: btw,  (5) may contain some typos.\n\n3) In practice, how to balance the three parameters lambda_p, lambda_c, lambda_cur?\n\n ", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I did not assess the derivations or theory.", "review_assessment:_checking_correctness_of_experiments": "I did not assess the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper studies optimal control with low-dimensional representation.  The paper presents interesting progress, although I urge the authors to address all issues raised by reviewers in their revisions.
Decision: Accept (Poster)