We generalized a modified exponentialized estimator by pushing the robust-optimal (RO) index $\lambda$ to $-\infty$ for achieving robustness to outliers by optimizing a quasi-Minimin function. The robustness is realized and controlled adaptively by the RO index without any predefined threshold. Optimality is guaranteed by expansion of the convexity region in the Hessian matrix to largely avoid local optima. Detailed quantitative analysis on both robustness and optimality are provided. The results of proposed experiments on fitting tasks for three noisy non-convex functions and the digits recognition task on the MNIST dataset consolidate the conclusions.

This paper aims to overcome a fundamental problem in the theory and application of deep neural networks (DNNs). We propose a method to solve the local minimum problem in training DNNs directly. Our method is based on the cross-entropy loss criterion's convexification by transforming the cross-entropy loss into a risk averting error (RAE) criterion. To alleviate numerical difficulties, a normalized RAE (NRAE) is employed. The convexity region of the cross-entropy loss expands as its risk sensitivity index (RSI) increases. Making the best use of the convexity region, our method starts training with an extensive RSI, gradually reduces it, and switches to the RAE as soon as the RAE is numerically feasible. After training converges, the resultant deep learning machine is expected to be inside the attraction basin of a global minimum of the cross-entropy loss. Numerical results are provided to show the effectiveness of the proposed method.

This paper proposes a set of new error criteria and learning approaches, Adaptive Normalized Risk-Averting Training (ANRAT), to attack the non-convex optimization problem in training deep neural networks (DNNs). Theoretically, we demonstrate its effectiveness on global and local convexity lower-bounded by the standard $L_p$-norm error. By analyzing the gradient on the convexity index $\lambda$, we explain the reason why to learn $\lambda$ adaptively using gradient descent works. In practice, we show how this method improves training of deep neural networks to solve visual recognition tasks on the MNIST and CIFAR-10 datasets. Without using pretraining or other tricks, we obtain results comparable or superior to those reported in recent literature on the same tasks using standard ConvNets + MSE/cross entropy. Performance on deep/shallow multilayer perceptrons and Denoised Auto-encoders is also explored. ANRAT can be combined with other quasi-Newton training methods, innovative network variants, regularization techniques and other specific tricks in DNNs. Other than unsupervised pretraining, it provides a new perspective to address the non-convex optimization problem in DNNs.

This paper proposes a set of new error criteria and a learning approach, called Adaptive Normalized Risk-Averting Training (ANRAT) to attack the non-convex optimization problem in training deep neural networks without pretraining. Theoretically, we demonstrate its effectiveness based on the expansion of the convexity region. By analyzing the gradient on the convexity index $\lambda$, we explain the reason why our learning method using gradient descent works. In practice, we show how this training method is successfully applied for improved training of deep neural networks to solve visual recognition tasks on the MNIST and CIFAR-10 datasets. Using simple experimental settings without pretraining and other tricks, we obtain results comparable or superior to those reported in recent literature on the same tasks using standard ConvNets + MSE/cross entropy. Performance on deep/shallow multilayer perceptron and Denoised Auto-encoder is also explored. ANRAT can be combined with other quasi-Newton training methods, innovative network variants, regularization techniques and other common tricks in DNNs. Other than unsupervised pretraining, it provides a new perspective to address the non-convex optimization strategy in training DNNs.

We propose a minimum distance estimation method for robust regression in sparse high-dimensional settings. The traditional likelihood-based estimators lack resilience against outliers, a critical issue when dealing with high-dimensional noisy data. Our method, Minimum Distance Lasso (MD-Lasso), combines minimum distance functionals, customarily used in nonparametric estimation for their robustness, with l1-regularization for high-dimensional regression. The geometry of MD-Lasso is key to its consistency and robustness. The estimator is governed by a scaling parameter that caps the influence of outliers: the loss per observation is locally convex and close to quadratic for small squared residuals, and flattens for squared residuals larger than the scaling parameter. As the parameter approaches infinity, the estimator becomes equivalent to least-squares Lasso. MD-Lasso enjoys fast convergence rates under mild conditions on the model error distribution, which hold for any of the solutions in a convexity region around the true parameter and in certain cases for every solution. Remarkably, a first-order optimization method is able to produce iterates very close to the consistent solutions, with geometric convergence and regardless of the initialization. A connection is established with re-weighted least-squares that intuitively explains MD-Lasso robustness. The merits of our method are demonstrated through simulation and eQTL data analysis.