
Submitted by Assigned_Reviewer_1
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
Summary
The paper presents a new method for path integral control. The proposed method leverages that the system control
matrix G is often known, and the uncontrolled dynamics (or dynamics under a reference controller) can be learned
using a Gaussian Process. The constraint on the reward function takes a more general shape than in previous PI
approaches which means among others that noise and controls can act in different subspaces. The authors also show
how their framework can be used for generalizing from known to new tasks, and evaluate the method on three simulated
robotics problems, where their method compares favourably to SOTA reinforcement learning and control methods.
Quality
Generally, the derivations seem correct and principled (although I'm unsure about the task generalization, see below).
Related work is sufficiently discussed, the authors point out the similarities and differences to related work and
compare to state of the art methods. The experiments are convincing, altough some details are missing (see below).
The task generalization seems odd: (13) states that the exponent of the reward functions will be linearly combined
to yield the exponent of the reward function of the new task. However, it's not clear to me how that results in the
exponent of the value function being the same kind of linear combination (14). In any case, this should be explained,
but I think that actually there might be an error here ((14) also states psi is the exponent of the cost function,
which holds only for the last time step as shown in the equation around line 247).
As defined by the authors, the task
generalization is specific to the case that the task is defined by a target, although it seems that other
task variations would be possible as long as a suitable task similarity kernel can be defined.
The experiments considers relevant and realistic tasks, and compare to SOTA methods. The technical details are a bit sparse
in the experimental section: the paper should mention what the dynamics and reward functions are, and how long sampled trajectories
are, in order for the experiment to be reproducible. (Possibly in the appendix). One baseline is iterative PI control,
is this one of the methods in table 1? It would be insightful to see how it compares on these qualitative aspects.
The comparison plots in Figures 1 and 2 should show error bars. The reported score is the exponent of the value fc, psi.
Is this the psi as calculated by the robot? What if the robot is overly optimistic in its calculation of psi? It would
be more objective to report the average (or cumulative (discounted)) reward (that is also what the algorithm sets out to optimize).
Clarity
Generally, the paper is wellwritten and explains the proposed methods rather well. The comparison to other methods, e.g.
in Table 1, helps understanding the relationship to the SOTA. There are a couple of grammatical and formatting errors,
see below under Minor points. One confusing point is that the GP has different regression targets in (2) (line 094) and line 208,
which doesn't include the reference controls. This should be made consistent or explained. The authors should explain how
the hyperparameters of the GP are set. It's unclear what's meant by "the posterior distribution can be obtained by
constraining the joint distribution"  is conditioning meant?
Algorithm 1 is confusing  here, it looks as if an openloop
control sequence is optimized while the rest of the paper discusses learning a feedback controller.
If it's really just an openloop sequence u_1 ... u_T that's returned in line 13 of the algorith, how can the algorithm deal
with the noise? This should
be clarified or corrected.
Originality and Significance
Although the main ingredients of the proposed algorithm have been used in earlier algorithms (propagation through an learned
stochastic forward model as in [15], path integral control), the algorithm combines these in a novel way. As far as I know,
this is an original formulation that seems to attain very good performance. By evaluating on realistic problems against strong
benchmarks, the proposed algorithm can be concluded to be a significant improvement over prior work.
Comments on the rebuttal:
Thanks for clarifying (14). Still, it seems there is a typo and Psi should be Phi (or Psi_t+dt should be included in the middle part). To me, it's therefore unfortunately still unclear what's meant here. I changed my confidence
& score accordingly. Thanks for clarifying the openloop part, I would really change the notation here to avoid confusion.
I'm also still confused about what meant by the psi in figure (1), as I understand it this is the exponent of the final costs, but average cost would be more standard. Maybe the average cost could additionally be reported in the supplementary material?
Minor points
* Although generally wellwritten, the paper has some grammar issues. I recommend letting someone proofread it
(line 040 "has been existed", line 143 (while > in contrast ?), line 231  comma instead of full stop or reformulate,
line 345 is based on ... model > are based on ... models
)
* the authors should avoid creating empty sections (e.g. between sec 2 and sec 2.1)
* there are some formatting issues (margins in line 128, table 1, brackets not scaled correctly in (8))
* would be clearer if the authors state how to obtain \tilde{G} from G
* line 192 table.1. > Table 1.
* notation: in (13) x_t is written boldface on the righthand side but italic on the lefthand side.
* line 339: Pilco requires an optimizer for policy evaluation > is policy improvement meant?
* line 332 "6 actuators on each joint" > is "one actuator per joint" meant?
Q2: Please summarize your review in 12 sentences
The paper presents a new method for path integral control. The method is evaluated in convincing experiments. Possibly, there is an issue with the derivation for the multitask setting, I would like to see the author's reply on this point.
Submitted by Assigned_Reviewer_2
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
The authors propose a novel formulation of path integral control, where they use an exponential form of the value function, which allows them to solve the path integrals of the cost analytically, and in particular they can evaluate the path integrals backwards. This gives them the ability to optimize using a forwardbackward approach where they use a GP to model the paths forward in time, and then run the backward integration to get the path costs and derivatives, and then use a numerical technique to update the controls.
Quality: The quality of the paper is relatively high, in technical correctness. I enjoyed reading the paper and learned something new.
I was surprised that this technique outperformed iterative PI  unless I am missing something, this is almost certainly a problem with the number of samples (10^3 for cartpole, 10^4 for double pendulum) which seem relatively low. No doubt the new technique is more efficient, but the authors need to be careful with their baseline evaluation.
Clarity: The major comment I have about the paper is that the authors move quickly through some key steps. In particular, the move from eqaution 7 to 8 is nontrivial, and is key to the overall paper. Similarly, the moves from the initial integral of \Psi_t (midpage 5) to the final recursive form of \Phi_{t+dt)\Psi_{t+dt} is even harder.
Originality: The use of a GP for the model is not new, but the forwardbackward is, additionally the ability to use the GP to infer the integral for a changed value function is at least new to me.
Significance: An interesting and substantial improvement to existing path integral methods.
Please number all equations  I understand there is a school of thought that equations should only be numbered if they are referenced in the text, but that stylistic conceit creates major problems when others want to refer to the equations outside the text of the paper, either in reviews (like this one), in other papers referencing this one, or in discussion.
Q2: Please summarize your review in 12 sentences
The authors have proposed a novel method that advances the state of the art. Overall a good paper, although not easy to read in places. This is largely due to page limitations so I can't hold the authors too much to blame for this.
Submitted by Assigned_Reviewer_3
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http://nips.cc/PaperInformation/ReviewerInstructions)
The paper presents a novel PIrelated method for stochastic optimal control. The main idea is to use a Gaussian process to model the trajectories resulting of applying a proposal control $u^old$ on the uncontrolled dynamics. The proposed Bellman equation is defined to optimize for $\delta u$, the control update at each iteration. The method takes advantage of analytical methods for GP inference to update the GP parameters efficiently and it is evaluated experimentally in three simulated tasks.
The idea is interesting and authors seem to have a working method that can compete with existing ones. I like the paper in general, but I think the following points need to be addressed:
The problem formulation section is a bit confusing, since there are two optimizations involved: one is the optimization over $u$ to find the optimal control and the other is the one that optimizes $\delta u$. From 2.1, it seems that the optimal u is given together with a proposal control u^old, which is clearly not the case, since finding the optimal u is exactly the main task. Authors need to clarify this point.
The organization of the paper could be significantly improved: since the proposed approach is not introduced in 3.1 but earlier, I would move subsection 3.1 after subsection 2.1. This would also improve the readability, since Eq (9) involves the inference step explained in subsection 3.1.
I also found confusing section 3.1, in which the GP is defined (line 208) for the uncontrolled process (u=0) which contradicts the definition in Eq(2) which depends on the current proposal control. Additionally, if the GP is initialized using sampled data from the uncontrolled process, authors should also clarify how the poor conditioning of the uncontrolled process in terms of effective sample size affects this initialization.
Also related to how the GP model is updated, it looks like the formulation ignores the cost of the current control sequence, see Eq(5). In iterative PI control this term enters in the equation as a RadonNikodim derivative [16,17]. Can the authors point at the equivalent term in their formulation? Can we still talk about a generalization of PI control or is this another policy search, PIrelated method?
The use of belief propagation for forward inference may fail when the dynamics around the reference change abruptly, i.e. they are nonGaussian. This happens in the presence of obstacles, for example, which does not seem to occur in the presented experiments. Authors should clarify if this is a potential problem or not.
About notation, although authors present the method as modelbased, it is not clear that a model of the dynamics is used (they only have a probability distribution over trajectories via the GP). I suggest different notation for clarity.
Authors do a good job relating their approach with existing PIrelated methods, a difficult task given the (already large) current literature on PI control. I have the following remarks in that respect:
 In PI^2 [8] the constraint is not ignored, but PI^2 is an openloop policy search method, so the constraint is meaningless and the same argumentation of PI^2CMA holds.  Feedback PI [17] uses a parameterized feedback term (the second order correction), but not a parameterized policy.  PIREPS [11] is modelbased (although the model can be learned, as in guided policy search [23])  Iterative PI [16] is original PI with importance sampling and therefore: modelbased, with same structural constraint as PI and without parameterized policy.
These remarks do not involve table 1 only, but text spread through the whole paper.
minor:
line 120: minimize(s) lines 144145: sentence unclear: "while (...)." line 146: (to) act line 229: based on (a) probabilistic
Q2: Please summarize your review in 12 sentences
The paper presents a novel PIrelated method for stochastic optimal control that uses a Gaussian process to model trajectories. The approach is promising but it is not clear if this is a generalization of PI control or it is a policy search, PIrelated method. Presentation should be improved and some other points need to be clarified.
Q1:Author
rebuttal: Please respond to any concerns raised in the reviews. There are
no constraints on how you want to argue your case, except for the fact
that your text should be limited to a maximum of 5000 characters. Note
however, that reviewers and area chairs are busy and may not read long
vague rebuttals. It is in your own interest to be concise and to the
point.
We thank all reviewers for their constructive
comments. We will improve the manuscript as
suggested.
Reviewer_1 As the reviewer pointed out, it is
straightforward to see eq.(14) holds for the final time step. Since (14)
is the solution to a linearized HJB PDE (7), if the solution holds on the
boundary (which is the final time step) then it holds everywhere.
Moreover, the linear recursive form of \Psi (line 244252) also leads to
linear form in (14) from t to T. However for any nonlinear HJB PDE this is
not the case. In our particular case this linear combination is valid. We
will clarify this point. The iterative PI used as the baseline is the
one in [16] under known dynamics. The standard PI in table 1 provides a
theoretical framework but is not directly applicable to highly
underactuated systems such as DPC in our experiment. The optimized
control sequence in algorithm 1 is not an openloop sequence. Each
controller is a function of the current state/future beliefs (eq.9). We
will clarify this.
Reviewer_2 The iterative PI is a
samplingbased approach that relies on random explorations under known
dynamics. It requires sampling at each iteration. From (7) to (8) we
applied the FeynmanKac formula. We will reorganize some of the
derivations. We will number all equations.
Reviewer_3 In our
iterative scheme we solve a control problem at each iteration by
optimizing \delta u (Bellman eq. 5). Mathematically our problem is
defined differently from a standard optimal control problem. We will
further clarify this. In regard to the organization. Sec.2 introduces a
theoretical framework and in Sec.3 we design an algorithm. But we will
take into account the reviewer's opinions on the reorganization. Thanks
for pointing out line 208, we will fix the typo. Regarding the control
cost in our iterative scheme, the reviewer has a valid point, but it
appears that there is a misunderstanding. Our iterative control problem is
defined differently from the iterative PI based on importance sampling. At
each iteration the control cost matrix is adapted based on uncertainty of
the dynamics. While the iterative PI solves the same control problem with
a fixed criterion by iteratively sampling from different distributions.
Our iterative scheme is not a generalization of importance sampling. It is
more general since we solve a HJB PDE under uncertain dynamics. We will
clarify based on your suggestions. Our method is based on the
assumption of Gaussian systems (line 9293). Therefore we did not use
nonGaussian dynamics in our experiments. We agree with the reviewer that
for nonGaussian systems there may be a potential problem. We will discuss
this issue in the final version. The proposed method is modelbased
since we model the transition probabilities of the dynamics using GP
inferences. We will reconsider the notations.
Reviewer_4 The
reviewer stated the main advantage of PI is obtaining optimal policy
without approximations. We would like to remind that exact solution to PI
control exists for systems for which the Backward ChapmanKolmogorov PDE
has an analytical and therefore exact solution. For general nonlinear
systems the samplingbased policy is an approximation up to order of dt^2.
However for uncertain dynamics the case is different. In our method, the
path integral (8) and its gradient can be computed analytically thanks to
the GP representations of the transition probabilities, this lead to
explicit feedback control laws. In regard to comparing our method with
"algorithms using approximations such as iLQGbased approaches". The PDDP
algorithm [18] used for comparison is one of the aforementioned
algorithms. We mentioned in line 343345 that we did not numerically
compare our method with S.Levine's work because of different model
learning schemes. Our method and the methods for comparison (PILCO, PDDP)
use GPs to learn dynamics, therefore the performance differences are due
to the controller designs rather than the learned models. The reviewer
suggested that we have to show the computation speed of our method is
significantly faster than sampling based approaches. There seems to be a
misunderstanding. The point of this method is "sample efficiency" under
uncertain dynamics. We did not claim that our method is faster than other
samplingbased approaches. There are definitely faster methods but need
more samples. The two methods we used for comparison are the
stateoftheart in terms of sample efficiency. In addition we do not
sample from GPs. It is known that sampling functions from GPs could be
very challenging (significantly different from sampling random variables).
We will work on further comparisons with methods such as PIREPS in an
extended version of this work. We mentioned in line 321 that the
composite control law is different from an interpolation. Detailed
discussions can be found in Todorov's work [22].
Reviewer_5 and
6 We thank the reviewers for their comments. 
