Mokhtari, Aryan, Ozdaglar, Asuman, Jadbabaie, Ali

In this paper, we study the problem of escaping from saddle points in smooth nonconvex optimization problems subject to a convex set $\mathcal{C}$. We propose a generic framework that yields convergence to a second-order stationary point of the problem, if the convex set $\mathcal{C}$ is simple for a quadratic objective function. Specifically, our results hold if one can find a $\rho$-approximate solution of a quadratic program subject to $\mathcal{C}$ in polynomial time, where $\rho 1$ is a positive constant that depends on the structure of the set $\mathcal{C}$. Under this condition, we show that the sequence of iterates generated by the proposed framework reaches an $(\epsilon,\gamma)$-second order stationary point (SOSP) in at most $\mathcal{O}(\max\{\epsilon {-2},\rho {-3}\gamma {-3}\})$ iterations. We further characterize the overall complexity of reaching an SOSP when the convex set $\mathcal{C}$ can be written as a set of quadratic constraints and the objective function Hessian has a specific structure over the convex $\mathcal{C}$.

If we are doing a binary classification using logistic regression, we often use the cross entropy function as our loss function. Question: However, if we are doing linear regression, we often use squared-error as our loss function. Are there any specific reasons for using the cross entropy function instead of using squared-error or the classification error in logistic regression? I read somewhere that, if we use squared-error for binary classification, the resulting loss function would be non-convex. Is this the only reason reason, or is there any other deeper reason which I am missing?

In machine learning, you start by defining a task and a model. The model consists of an architecture and parameters. For a given architecture, the values of the parameters determine how accurately the model performs the task. But how do you find good values? By defining a loss function that evaluates how well the model performs.

In this post we will talk about applying gradient descent on \(m\) training examples. Now the question is how we can define what gradient descent is? A gradient descent is an efficient optimization algorithm that attempts to find a global minimum of a function. It also enables a model to calculate the gradient or direction that the model should take to reduce errors (differences between actual \(y\) and predicted \(\hat{y}\)). Now let's remind ourselves what the cost function is?

Wei, Xiaohan, Yu, Hao, Ling, Qing, Neely, Michael

We propose a new primal-dual homotopy smoothing algorithm for a linearly constrained convex program, where neither the primal nor the dual function has to be smooth or strongly convex. The best known iteration complexity solving such a non-smooth problem is $\mathcal{O}(\varepsilon {-1})$. This result improves upon the $\mathcal{O}(\varepsilon {-1})$ convergence time bound achieved by existing distributed optimization algorithms. Simulation experiments also demonstrate the performance of our proposed algorithm. Papers published at the Neural Information Processing Systems Conference.