19 research outputs found
DRIP: Domain Refinement Iteration with Polytopes for Backward Reachability Analysis of Neural Feedback Loops
Safety certification of data-driven control techniques remains a major open
problem. This work investigates backward reachability as a framework for
providing collision avoidance guarantees for systems controlled by neural
network (NN) policies. Because NNs are typically not invertible, existing
methods conservatively assume a domain over which to relax the NN, which causes
loose over-approximations of the set of states that could lead the system into
the obstacle (i.e., backprojection (BP) sets). To address this issue, we
introduce DRIP, an algorithm with a refinement loop on the relaxation domain,
which substantially tightens the BP set bounds. Furthermore, we introduce a
formulation that enables directly obtaining closed-form representations of
polytopes to bound the BP sets tighter than prior work, which required solving
linear programs and using hyper-rectangles. Furthermore, this work extends the
NN relaxation algorithm to handle polytope domains, which further tightens the
bounds on BP sets. DRIP is demonstrated in numerical experiments on control
systems, including a ground robot controlled by a learned NN obstacle avoidance
policy
Adaptive Neural Compilation
This paper proposes an adaptive neural-compilation framework to address the
problem of efficient program learning. Traditional code optimisation strategies
used in compilers are based on applying pre-specified set of transformations
that make the code faster to execute without changing its semantics. In
contrast, our work involves adapting programs to make them more efficient while
considering correctness only on a target input distribution. Our approach is
inspired by the recent works on differentiable representations of programs. We
show that it is possible to compile programs written in a low-level language to
a differentiable representation. We also show how programs in this
representation can be optimised to make them efficient on a target distribution
of inputs. Experimental results demonstrate that our approach enables learning
specifically-tuned algorithms for given data distributions with a high success
rate.Comment: Submitted to NIPS 2016, code and supplementary materials will be
available on author's pag
A Unified View of Piecewise Linear Neural Network Verification
The success of Deep Learning and its potential use in many safety-critical
applications has motivated research on formal verification of Neural Network
(NN) models. Despite the reputation of learned NN models to behave as black
boxes and the theoretical hardness of proving their properties, researchers
have been successful in verifying some classes of models by exploiting their
piecewise linear structure and taking insights from formal methods such as
Satisifiability Modulo Theory. These methods are however still far from scaling
to realistic neural networks. To facilitate progress on this crucial area, we
make two key contributions. First, we present a unified framework that
encompasses previous methods. This analysis results in the identification of
new methods that combine the strengths of multiple existing approaches,
accomplishing a speedup of two orders of magnitude compared to the previous
state of the art. Second, we propose a new data set of benchmarks which
includes a collection of previously released testcases. We use the benchmark to
provide the first experimental comparison of existing algorithms and identify
the factors impacting the hardness of verification problems.Comment: Updated version of "Piecewise Linear Neural Network verification: A
comparative study
Efficient Linear Programming for Dense CRFs
The fully connected conditional random field (CRF) with Gaussian pairwise
potentials has proven popular and effective for multi-class semantic
segmentation. While the energy of a dense CRF can be minimized accurately using
a linear programming (LP) relaxation, the state-of-the-art algorithm is too
slow to be useful in practice. To alleviate this deficiency, we introduce an
efficient LP minimization algorithm for dense CRFs. To this end, we develop a
proximal minimization framework, where the dual of each proximal problem is
optimized via block coordinate descent. We show that each block of variables
can be efficiently optimized. Specifically, for one block, the problem
decomposes into significantly smaller subproblems, each of which is defined
over a single pixel. For the other block, the problem is optimized via
conditional gradient descent. This has two advantages: 1) the conditional
gradient can be computed in a time linear in the number of pixels and labels;
and 2) the optimal step size can be computed analytically. Our experiments on
standard datasets provide compelling evidence that our approach outperforms all
existing baselines including the previous LP based approach for dense CRFs.Comment: 24 pages, 10 figures and 4 table
Efficient Relaxations for Dense CRFs with Sparse Higher Order Potentials
Dense conditional random fields (CRFs) have become a popular framework for
modelling several problems in computer vision such as stereo correspondence and
multi-class semantic segmentation. By modelling long-range interactions, dense
CRFs provide a labelling that captures finer detail than their sparse
counterparts. Currently, the state-of-the-art algorithm performs mean-field
inference using a filter-based method but fails to provide a strong theoretical
guarantee on the quality of the solution. A question naturally arises as to
whether it is possible to obtain a maximum a posteriori (MAP) estimate of a
dense CRF using a principled method. Within this paper, we show that this is
indeed possible. We will show that, by using a filter-based method, continuous
relaxations of the MAP problem can be optimised efficiently using
state-of-the-art algorithms. Specifically, we will solve a quadratic
programming (QP) relaxation using the Frank-Wolfe algorithm and a linear
programming (LP) relaxation by developing a proximal minimisation framework. By
exploiting labelling consistency in the higher-order potentials and utilising
the filter-based method, we are able to formulate the above algorithms such
that each iteration has a complexity linear in the number of classes and random
variables. The presented algorithms can be applied to any labelling problem
using a dense CRF with sparse higher-order potentials. In this paper, we use
semantic segmentation as an example application as it demonstrates the ability
of the algorithm to scale to dense CRFs with large dimensions. We perform
experiments on the Pascal dataset to indicate that the presented algorithms are
able to attain lower energies than the mean-field inference method
Branch and Bound for Piecewise Linear Neural Network Verification
The success of Deep Learning and its potential use in many safety-critical
applications has motivated research on formal verification of Neural Network
(NN) models. In this context, verification involves proving or disproving that
an NN model satisfies certain input-output properties. Despite the reputation
of learned NN models as black boxes, and the theoretical hardness of proving
useful properties about them, researchers have been successful in verifying
some classes of models by exploiting their piecewise linear structure and
taking insights from formal methods such as Satisifiability Modulo Theory.
However, these methods are still far from scaling to realistic neural networks.
To facilitate progress on this crucial area, we exploit the Mixed Integer
Linear Programming (MIP) formulation of verification to propose a family of
algorithms based on Branch-and-Bound (BaB). We show that our family contains
previous verification methods as special cases. With the help of the BaB
framework, we make three key contributions. Firstly, we identify new methods
that combine the strengths of multiple existing approaches, accomplishing
significant performance improvements over previous state of the art. Secondly,
we introduce an effective branching strategy on ReLU non-linearities. This
branching strategy allows us to efficiently and successfully deal with high
input dimensional problems with convolutional network architecture, on which
previous methods fail frequently. Finally, we propose comprehensive test data
sets and benchmarks which includes a collection of previously released
testcases. We use the data sets to conduct a thorough experimental comparison
of existing and new algorithms and to provide an inclusive analysis of the
factors impacting the hardness of verification problems