You Shall Pass: Dealing with the Zero-Gradient Problem in Predict and Optimize for Convex Optimization
CoRR(2023)
摘要
Predict and optimize is an increasingly popular decision-making paradigm that
employs machine learning to predict unknown parameters of optimization
problems. Instead of minimizing the prediction error of the parameters, it
trains predictive models using task performance as a loss function. The key
challenge to train such models is the computation of the Jacobian of the
solution of the optimization problem with respect to its parameters. For linear
problems, this Jacobian is known to be zero or undefined; hence, approximations
are usually employed. For non-linear convex problems, however, it is common to
use the exact Jacobian. This paper demonstrates that the zero-gradient problem
appears in the non-linear case as well – the Jacobian can have a sizeable null
space, thereby causing the training process to get stuck in suboptimal points.
Through formal proofs, this paper shows that smoothing the feasible set
resolves this problem. Combining this insight with known techniques from the
literature, such as quadratic programming approximation and projection distance
regularization, a novel method to approximate the Jacobian is derived. In
simulation experiments, the proposed method increases the performance in the
non-linear case and at least matches the existing state-of-the-art methods for
linear problems.
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关键词
optimization,optimize
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