
Submitted by
Assigned_Reviewer_5
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 authors presented a recurrent neuronal
network, modeled as a continuoustime dynamical system, that can solve
constraint satisfaction problems.
Since the authors did not
completely define their recurrent neuronal network, it is hard to
understand at least for me. For example, v(t) and u(t) appear without
definition in line 107. The single WTA network described in line 151 and
Figure 1 (a) is not mathematically defined. Synaptic connections of the
coupled WTA circuit described in lines 158159 and Figure2 are not
defined. The authors just said that they asymmetrically coupled a number
of WTA circuits. I could not understand how the binary value is encoded as
the state of the oscillatory network regarding lines 178180. WA and WB
appear without definition in line 191. The variables X1 and X2 appear
without definition in line 199. Are X1 and X2 the synaptic weights or the
names of two WTA circuits?
It is well known that neural network
models tends to behave chaotic in the case that the synaptic connections
dynamically change, for example, the short term synaptic depression. I
suppose that that is the reason why the present model seems to be
stochastic even thought it is defined as the deterministic system.
I also think that the system size of the present is too small (ten
units with 9 constraint). For example, more than 10^3 units are usually
discussed in the 3SAT problem. I have to say that the example of the
present paper is not enough in the framework of the CSP problem
. Q2: Please summarize your review in 12 sentences
I could not understand contents of this paper, because
the description seems to be incomplete as described in 1.
Submitted by
Assigned_Reviewer_7
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 new approach for solving
constraint satisfaction problems (CSPs) in a neural architecture. While
previous attempts in this direction have largely relied on stochastic
neuron models to prevent a neural network from getting stuck in local
optima (e.g. through the use of Boltzmann machines), the proposed
architecture is capable of finding optimal solutions to problems using
purely deterministic network dynamics. This is achieved by coupling
oscillator modules such that the strength of their mutual interaction
depends on their phase difference. Although the paper does not provide a
rigorous theoretical explanation for it, these phase differences are
observed to vary irregularly in simulations, thus providing sufficient
exploratory drive to escape local optima. In small simulations of a CSP
with ten binary variables and nine tertiary constraints the network is
observed to find either of the two correct solutions (which satisfy all
constraints) in each trial. Interestingly, exploration is driven purely by
violated constraints and thus, in contrast to stochastic approaches to
solving CSPs, the network will stay in the global optimum once it is found
(without the need for morphing into a greedy search as, for example, in
simulated annealing). If no solution exists which satisfies all
constraints the network behavior resembles a sampling process from a Gibbs
distribution with a linear energy term. Unfortunately, a theoretical
explanation of this behavior could not be provided by the authors.
The presented approach to solving CSPs in a deterministic neural
architecture is, to the best of my knowledge, original and novel. It
provides both an innovative approach for solving CSPs in neuromorphic
hardware and a fresh perspective on oscillations and synchronization
between oscillating modules (and lack thereof in the form of irregularly
varying phase differences) in neuroscience.
The applications are
nontrivial, although too few and too small to provide insight into the
scalability of the method. On the positive side, the problems were chosen
to be both illustrative and difficult in the sense that the energy
landscape was designed to possess several local optima and more than one
global optimum (especially in Fig. 3). On the negative side, the state
spaces of the considered problems are very small (2^10 or 2^11). For such
state spaces simple exhaustive search works well and is a much easier
option. I would suspect that larger simulations would show that the
algorithm still performs well, within what one may expect from a parallel
iterative algorithm. Unfortunately the paper does not address scalability,
and so the utility of the approach for larger problems remains unclear. At
least a discussion of scalability is warranted and would be highly useful
for the understanding of the method.
In terms of clarity and
quality of figures and text the paper rates highly, yet no theoretical
arguments were given for the observed behavior of the network. The paper
is well organized, the model is for the most part clearly defined, the
chosen examples are illustrative, the figures convey a clear message, and
the explanations of the network dynamics are insightful. The lack of
theoretical arguments is an obvious shortcoming of the paper. Furthermore,
I'm missing a specification of the parameters used in the simulations
(parameters in Eq. (1) and (2)). Were the same parameters used in the
simulations for Fig 2, 3 & 4?
The paper is potentially
significant, as it provides a new and relatively simple neural
architecture for solving CSPs with deterministic neurons and has a large
potential for neuromorphic applications. Furthermore, from the perspective
of neuroscience, the potential relation between oscillations and the
search for “consistent states” in the brain is intriguing and certainly
worth pursuing.
Q2: Please summarize your review in
12 sentences
This is an interesting and well written paper that
provides an innovative model for solving CSPs in neural networks with
oscillations, offering potential applications in neuromorphic engineering
and a new perspective on oscillations in neuroscience. Two shortcomings of
the paper are the lack of a theoretical treatment, and the lack of
simulations concerning (or at least a discussion of)
scalability. Submitted by
Assigned_Reviewer_8
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)
Overview:
The paper studies a neurodynamic
system that is able to solve constraint satisfaction problems (CSP).
The dynamic equations for neural interaction and synaptic
weightchanges are relatively straightforward (equations 1 and 2),
the archictectures of the networks are relatively complex. The
elementary computational units Are winnertakeall (WTA) type networks
that oscillate. It is stated that a crucial property of the elementary
circuits is their sensitivity to input only at specific points of
their oscillatory period(s). If the connectivity between such WTA
units is chosen to reflect a CSP, it is stated that the dynamics
finally settles into a state that represents a solution to the CSP.
Simulations of the neurodynamics for small and intermediately large
CSPs empirically show that the neurodynamics can find solutions, and
that it can find approximately optimal solutions if there are no fully
consistent variable assignments.
Evaluation:
The
paper is generally well written and the main points become clear. The
question what computations neural circuits perform and how they are
performed is certainly important and interesting. There is also
evidence for WTA mechanisms that implement a form of possible
competition between neural untis. Also the effectiveness of input
during certain points of the phase is an interesting property because
a small input differences can be amplified (and response can be
normalized). Even if weakly coupled to an input, such networks can be
very sensitive.
While the general setup is clear, I was missing a
more extensive discussion about the parameters of the network. Usually
such parameters have to be tuned. It does not become clear how
difficult this is for the suggested system. Also a discussion of
related neurodynamic approaches is missing. Neural dynamics that
become sensitive to input in a critical period of their oscillation have
been described before, sometimes in the form of sensitivity to initial
states. Such networks all have a WTA characteristics and at least some
should be discussed (see, e.g., work by O'Reilly, 2001; Spratling and
Johnson, 2002; Lucke, Malsburg, 2004). The difference to the here
discussed system is the goal: here it is finding solutions to CSPs,
the other networks address component finding in data. Finding CSP
solutions is a well accepted problem in computer science but it
becomes much less clear why biological networks should be optimized
for such a problem  please discuss. Still it is interesting from a
technical point, and I agree that implementations of inherently
parallel neurodynamic approaches via analog VLSI are very
interesting. In this respect please discuss two questions, however:
(1) isn't the complexity of the problem mapped into the complexity of
the network architecture? (2) can one be confident that the problem
solving dynamics do not break down if the problem sizes are scaled up?
What about the total number of variables?
The behavior of the
network is interesting including the behavior for not fully
satisfiable problems. The empirical finding of exponential scaling is
not uninteresting but the sampling analogy is taken too far for my
taste. The energy function could also be used for the satisfiable
problem (Sec. 3.1). But if the network remains in a solution there, it
is not spending time proportional to exp(cE). Additionally, the
authors themselves say that the network will probably not explore
highenergy states. So there is only a coarse analogy here as far as I
can see.
Also the criticism of other approaches is too general. Of
course there are many approaches on the technical side that have
similar properties. Also it is not clear how computationally costly
the mechanisms to avoid local optima really are for the studied
dynamics  time to find optimal solutions may scale exponentially with
problem size  or the architecture does.
In response to author
feedback:
The feedback has clarified most of my questions. I agree
that scalability will be exponential in the worst case. What would be
interesting, though, is how the average time to find a solution scales
with problem size. This should be a point for the revised discussion.
I also assume that more care is taken regarding the too broad
criticism of other approaches, that it is discussed why CSP solving is
interesting for biological systems and about the analogy to sampling.
Given these improvements, I remain happy with my initial score.
Q2: Please summarize your review in 12
sentences
Interesting neural dynamic approach to solve
constraint satisfaction problems. May ultimately result in
stateoftheart approaches in terms of energy cost per CSP problem
solution if implemented on VLSI. Submitted by
Assigned_Reviewer_9
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 authors propose a neural network model
whose deterministic dynamics implement a search and ultimately global
solution to a constraint satisfaction problem (CSP) when a solution
exists. The fundamental computational unit of the network is a
multistable oscillator that represent a binary variable. Multiple network
modules are connected to represent multiple variables, and additional
variables are defined as operations on these variables. When the modules
are connected in such a way to implement a constraint (a desired operation
on the variables), if the network dynamics are such that the constraint is
violated, the the phase difference between different oscillators
(variables) shifts such as to alter the certain state variables which
cause the constraint violation. Once the relevant state variables have
flipped, synaptic plasticity will cause this new state to become stable
and persist, leading to a network state that is consistent with all of the
desired constraints. When large simulations were conducted, the phase
relationship between the oscillators was seen to vary irregularly, which
resulted in sufficient exploratory drive for the network to escape local
optima and search for solutions consistent with all of the desired
constraints. With this property, the network has no need for often
employed stochastic neuron models to escape local optima. Simulations were
run showing that the network can find one of two globally optimal
solutions to a 10 variable, 9 constraint problem.
Review: The
paper is generally clear and well written. The figures are mostly
wellchosen (although figure 4 doesn't seem to add much) and support the
text. Network simulation examples are illustrative and provide the reader
with insight as to how the network elements operate and how the
oscillatory dynamics give rise to the plasticity which leads to a state
that satisfies all the constraints. To the best of my knowledge, this
approach is novel and the work posits a unique role for network
oscillations. Furthermore, the model has the desirable property that,
since search for a consistent solution is driven purely by violated
constraints, external input can be evaluated by the network in realtime
without the need for the system to switch between exploratory and greedy
optimization modes. This makes the proposed model relevant to biological
neural networks, which can rapidly assimilate external input with internal
network state without global parameter alterations.
The largest
shortcoming of the paper is that no explanation is provided as to why the
network exhibits the exploratory phase varying that gives rise to its CSP
solving ability for large network simulations. Considering the fact that
the main result of the paper is a direct result of this network phenomenon
some discussion of the origin of this phase variability would drastically
improve the submission. Section 3.2 only weakly supports the main results
of the paper and could be omitted to make room for a more elaborate
discussion or analysis of the phase variability. Additionally, no mention
is made as to the sensitivity of the network to parameters. A priori, most
network models require careful parameter tuning so that the network
dynamics are stable and exhibit a desired property. I imagine the same
would be true for the proposed model, specifically to achieve the state of
spontaneous phase shifting responsible for the network explorations.
Alternatively, perhaps the rich, phase varying dynamics are relatively
easy to achieve without careful parameter tuning due to the large size of
the network, similar to a chaotic neural network state, in which case such
a result would itself be interesting. Lastly, the scale of examples solved
by the network although useful to both illustrate how the network
operates and to prove that it can solve CSPs do not provide any insight
into how the network's ability to solve CSPs scales with problem size. A
more complete scaling analysis would be helpful to understand how the
proposed network would perform on larger optimizations, or if the
oscillatory dynamics would even perform the same function at a much larger
scale. Q2: Please summarize your review in 12
sentences
This is an interesting, original model for solving
CSPs in neural networks which employs a novel function for oscillatory
dynamics. A rigorous analysis of the origin of the phase variability that
gives rise to the network's ability to solve CSPs as well as the network's
parameter sensitivity would greatly improve the paper.
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 6000 characters. Note
however that reviewers and area chairs are very busy and may not read long
vague rebuttals. It is in your own interest to be concise and to the
point.
We thank the reviewers for their comments.
Regarding the issues raised by reviewer_5: 1) "v(t) and u(t)
appear without definition in line 107." v(t) and u(t) are defined in
the text directly following the equation in line 107, i.e. in lines
140  142
2) "The single WTA network described in line 151 and
Figure 1 (a) is not mathematically defined." The WTA network from
Figure 1(a) is well described in [15], which is the source given for the
paragraph in lines 82  90. The mathematical definition further follows
directly from the definition of its components, given in equation (1). All
synaptic weights within a single WTA are static
3)"Synaptic
connections of the coupled WTA circuit described in lines 158159 and
Figure2 are not defined" The synaptic connections in figure 2(a)
follow the same notation as those in figure 1(b): Black lines indicate
fixed weight connections. Orange lines indicate plastic connections that
follow the plasticity rule given in equation (2). All synaptic connections
are shown in Fig. 1b and Fig. 2a. In Fig. 2a, synaptic connections within
the same variable/oscillator are shown in light colors to avoid cluttering
the figure.
4)” I could NOT understand how the binary value is
encoded” The paragraph starting at line 177 describes how the value of
the binary variable is encoded. We mention in several places that variable
values are encoded in the identity of the limit cycle attractor chosen by
the network representing the variable.
5) "WA and WB appear
without definition in line 191" WA and WB are introduced in figure
1(b) and (d) (ca. line 110) and equation (2). They are the strengths of
two plastic weights. their effect is further described in lines 180  183.
6) "The variables X1 and X2 appear without definition in line 199"
In line 198 we refer the reader to figure 2(a) that illustrates X1 and
X2. X1 and X2 are just generic variables and we mention in line 196 that a
variable is represented by the network in Fig. 1b
Regarding the
comments of the other three reviewers
7)”state spaces of the
considered problems are very small”. In the problem in Section 3.1, an
intermediate variable with cardinality 4 was introduced for each
constraint. See Fig. 2c. The value of this intermediate variable can
become inconsistent with the value of the 3 problem variables connected to
it. There are thus 2^10*4^9 or 268 million possible states for the
variables/oscillators. This is the actual size of the space explored
by the network
8)Scalability issues: CSPs are in general NPhard.
Having the variables in the problem update their values concurrently like
in the proposed network does not change the fact that time to solution
will grow (in the worst case) exponentially with problem size. The size of
the network grows linearly.
9)”Lack of theoretical treatment”,
“origin of phase variability”: We recognize this as a shortcoming of
the paper. A work similar to ours, [Hopfield and Tank, 1985], does not
provide theoretical arguments for why the proposed Hopfield network is
able to escape local optima to find good solutions to the travelling
salesman problem, yet this work had significant impact. Many complex
systems are analyzed using simulations, and we have provided a qualitative
explanation for why the network behaves as it does. Phase variability is
an emergent feature of the dynamics, similar to chaos. Like chaos in
complex systems which in many cases is established using numerical
simulations, irregularity in phase relations is a phenomenon that we
observe in simulations. We are currently in the process of studying
analytically the phase variability and plan to describe our preliminary
results directly at NIPS and eventually in future publications.
 