**Session title**: Recent Advances in Statistical Learning

**Organizer**: Ming Yuan (Columbia)

**Chair:** Dong Xia (UCSD)

**Time:** June 4^{th}, 11:00am-12:30pm

**Location: **VEC 404

**Speech 1: Geometry of Optimization Landscapes and Implicit Regularization of Optimization Algorithms.**

**Speaker: Jason Lee (USC)**

**Abstract:**

We first study the problem of learning a Gaussian input two-layer ReLU network with positive output layer and and the symmetric matrix completion problem. Despite the non-convexity of both problems, we prove that every local minimizer is a global minimizer. Since gradient descent converges to local minimizers, this shows that simple gradient-based methods can find the global optimum of these non-convex problems.

In the second part, we analyze the implicit regularization effects of various optimization algorithms. In particular we prove that for least squares with mirror descent, the algorithm converges to the closest solution in terms of the bregman divergence. For linearly separable classification problems, we prove that the steepest descent with respect to a norm solves SVM with respect to the same norm. For over-parametrized non-convex problems such as matrix sensing or neural net with quadratic activation, we prove that gradient descent converges to the minimum nuclear norm solution, which allows for both meaningful optimization and generalization guarantees.

This is joint work with Rong Ge, Suriya Gunasekar, Tengyu Ma, Mor Shpigel, Daniel Soudry, and Nati Srebro.

**Speech 2: M-estimation with the Trimmed L1 penalty
Speaker: Aurelie Lozano (IBM)
Abstract:**

We study high-dimensional M-estimators with the trimmed L1 penalty. While standard L1 penalty incurs bias (shrinkage), trimmed L1 leaves the h largest entries penalty-free. This family of estimators include the Trimmed Lasso for sparse linear regression and its counterpart for sparse graphical model estimation. The trimmed L1 penalty is non-convex, but unlike other non-convex regularizers such as SCAD and MCP, it is not amenable and therefore prior analyzes cannot be applied.

We characterize the support recovery of the estimates as a function of the trimming parameter h. Under certain conditions, we show that for any local optimum, (i) if the trimming parameter h is smaller than the true support size, all zero entries of the true parameter vector are successfully estimated as zero, and (ii) if h is larger than the true support size, the non-relevant parameters of the local optimum have smaller absolute values than relevant parameters and hence relevant parameters are not penalized. We then bound the L2 error of any local optimum. These bounds are asymptotically comparable to those for non-convex amenable penalties such as SCAD or MCP, but enjoy better constants. We specialize our main results to linear regression and graphical model estimation.

Finally, we develop a fast provably convergent optimization algorithm for the trimmed regularizer problem. The algorithm has the same rate of convergence as difference of convex (DC)-based approaches, but is faster in practice and finds better objective values than recently proposed algorithms for DC optimization. Empirical results further demonstrate the value of L1 trimming.

**Speech 3: Methods of network comparison**

**Speaker: Sofia Olhede (UCL)**

**Abstract: **The topology of any complex system is key to understanding its structure and function. Fundamentally, algebraic topology guarantees that any system represented by a network can be understood through its closed paths. The length of each path provides a notion of scale, which is vitally important in characterizing dominant modes of system behavior. Here, by combining topology with scale, we prove the existence of universal features which reveal the dominant scales of any network. We use these features to compare several canonical network types in the context of a social media discussion which evolves through the sharing of rumors, leaks and other news. Our analysis enables for the first time a universal understanding of the balance between loops and tree-like structure across network scales, and an assessment of how this balance interacts with the spreading of information online. Crucially, our results allow networks to be quantified and compared in a purely model-free way that is theoretically sound, fully automated, and inherently scalable. This work is joint with Patrick Wolfe.