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1 Introduction
Deep learning has led to tremendous progress in machine learning in the last few years achieving state-of-the-art performance on complex image classification and speech recognition tasks krizhevsky2012imagenet. However, this progress has been tainted by disturbing revelations that state of the art networks can easily be fooled by making seemingly innocuous modifications to the input data that cause the network to change its prediction significantly szegedy2013intriguing; kurakin2016adversarial. While modifications to neural network training algorithms have been proposed to mitigate these effects, a comprehensive solution has remained elusive.
Further, neural networks are gaining widespread adoption, including in domains with critical safety constraints marston2015acas; DC; SD.
Given these factors, verification of neural networks has gained significant attention in recent research kolter2017provable; bunel2017piecewise add more
Most verification methods to date have been limited to piecewise linear neural networks. However, practical state-of-the-art performing neural networks have significant nonlinearities besides piecewise linear. In this report, we describe a general approach to verifying neural networks with arbitrary transfer functions.
2 Formulation
We will start with a layer-wise description of a neural networks
| (1a) | |||
where
is the vector of neural activations at layer
, is the pre-nonlinearity activations and is a component-wise nonlinearity.We use the notation to denote the -th component of the vectors and the -th component of the function , so that
Note that we do not necessarily need to assume that is the same for each
(so we can have layers where some of the neurons have tanh transfer functions while others have ReLUs and yet others have sigmoids).
Most verification problems can be posed as follows:
| (2a) | |||
| Subject to | (2b) | ||
| (2c) | |||
| (2d) | |||
| (2e) | |||
| (2f) | |||
| (2g) | |||
where is a set of constraints on the input (assumed to be convex) and are bounds on the pre and post nonlinear activations at each layer (that are inferred from the constraints on ). We assume for now that these bounds are given, but we later show how they can be inferred as well at a marginally small computational cost.
A concrete instance of a verification problem posed in this form would be when and and which corresponds to the search for an adversarial exmaple that causes the maximum deviation in the output of the network subject to the constraint that the input to the network does not change from a nominal value by more than in some norm.
We can bound the optimal value of (2) using the dual program:
| (3a) | |||
| Subject to | (3b) | ||
| (3c) | |||
| (3d) | |||
| (3e) | |||
| (3f) | |||
By weak duality, for any choice of , the above optimization problem provides a valid upper bound on the optimal value of (2).
We now look at solving the above optimization problem. Since the objective and constraints are separable in the layers, the variables in each layer can be optimized independently. For , we have
λ_l-1-W_l^Tμ_l)^Tx_l - (b_l)^Tμ_l which can also be solved trivially by setting each component of to its upper or lower bound. Finally, we have
where .
Since the objective is separable, one can solve separately for each component of :
This is a one-dimensional optimization problem and can be solved easily for most common transfer functions by simply looking at all the stationary points of the objective within the constraints plus the upper/lower bounds, and choosing among those the point at which the objective is largest. For most common transfer functions, since they are convex below and concave above (sigmoid, tanh all fall into this class), there are at most two stationary points within the domain, and hence the number of possibilities that need to be considered for this optimization is at most .
Finally, we need to solve
which can also be solved easily typically if is simply a norm ball (the solution would be of the form where is chosen such that is on the surface of the norm ball).
Once these problems are solved, we can construct the dual optimization problem:
| (4a) | |||
This optimization can be solved via a sub-gradient descent on . If the optimal are such that the objective of (3) is concave, then it can be guaranteed that there is no duality gap and the dual bound exactly matches the optimal value (2)
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