On the Stability Properties and the Optimization Landscape of Training Problems with Squared Loss for Neural Networks and General Nonlinear Conic Approximation Schemes
This addresses fundamental challenges in machine learning optimization for researchers and practitioners, but it is incremental as it builds on existing stability and landscape analyses.
The paper tackles the problem of instability and poor optimization landscapes in training neural networks and nonlinear conic approximation schemes with squared loss, showing that these issues arise from expressiveness and unrealizable labels, leading to discontinuous solution sets and spurious local minima that cannot be fixed by regularization.
We study the optimization landscape and the stability properties of training problems with squared loss for neural networks and general nonlinear conic approximation schemes. It is demonstrated that, if a nonlinear conic approximation scheme is considered that is (in an appropriately defined sense) more expressive than a classical linear approximation approach and if there exist unrealizable label vectors, then a training problem with squared loss is necessarily unstable in the sense that its solution set depends discontinuously on the label vector in the training data. We further prove that the same effects that are responsible for these instability properties are also the reason for the emergence of saddle points and spurious local minima, which may be arbitrarily far away from global solutions, and that neither the instability of the training problem nor the existence of spurious local minima can, in general, be overcome by adding a regularization term to the objective function that penalizes the size of the parameters in the approximation scheme. The latter results are shown to be true regardless of whether the assumption of realizability is satisfied or not. We demonstrate that our analysis in particular applies to training problems for free-knot interpolation schemes and deep and shallow neural networks with variable widths that involve an arbitrary mixture of various activation functions (e.g., binary, sigmoid, tanh, arctan, soft-sign, ISRU, soft-clip, SQNL, ReLU, leaky ReLU, soft-plus, bent identity, SILU, ISRLU, and ELU). In summary, the findings of this paper illustrate that the improved approximation properties of neural networks and general nonlinear conic approximation instruments are linked in a direct and quantifiable way to undesirable properties of the optimization problems that have to be solved in order to train them.