no code implementations • 19 Feb 2019 • Sai Ganesh Nagarajan, Ioannis Panageas
Moreover, for $d>1$ we show EM almost surely converges to the true mean for any measurable set $S$ when the map of EM has only three fixed points, namely $-\vec{\mu}, \vec{0}, \vec{\mu}$ (covariance matrix $\vec{\Sigma}$ is known), and prove local convergence if there are more than three fixed points.
no code implementations • ICLR 2020 • Vaggos Chatziafratis, Sai Ganesh Nagarajan, Ioannis Panageas, Xiao Wang
Motivated by our observation that the triangle waves used in Telgarsky's work contain points of period 3 - a period that is special in that it implies chaotic behavior based on the celebrated result by Li-Yorke - we proceed to give general lower bounds for the width needed to represent periodic functions as a function of the depth.
no code implementations • 17 Feb 2020 • Qi Lei, Sai Ganesh Nagarajan, Ioannis Panageas, Xiao Wang
In a recent series of papers it has been established that variants of Gradient Descent/Ascent and Mirror Descent exhibit last iterate convergence in convex-concave zero-sum games.
no code implementations • ICML 2020 • Vaggos Chatziafratis, Sai Ganesh Nagarajan, Ioannis Panageas
The expressivity of neural networks as a function of their depth, width and type of activation units has been an important question in deep learning theory.
no code implementations • 17 Jun 2020 • Arnab Bhattacharyya, Rathin Desai, Sai Ganesh Nagarajan, Ioannis Panageas
We show that ${\mu}$ and ${\Sigma}$ can be estimated with error $\epsilon$ in the Frobenius norm, using $\tilde{O}\left(\frac{\textrm{nz}({\Sigma}^{-1})}{\epsilon^2}\right)$ samples from a truncated $\mathcal{N}({\mu},{\Sigma})$ and having access to a membership oracle for $S$.
no code implementations • 20 Oct 2023 • Etienne Bamas, Sai Ganesh Nagarajan, Ola Svensson
For any $\alpha>2$, we show that $D^\alpha$ seeding guarantees in expectation an approximation factor of $$ O_\alpha \left((g_\alpha)^{2/\alpha}\cdot \left(\frac{\sigma_{\mathrm{max}}}{\sigma_{\mathrm{min}}}\right)^{2-4/\alpha}\cdot (\min\{\ell,\log k\})^{2/\alpha}\right)$$ with respect to the standard $k$-means cost of any underlying clustering; where $g_\alpha$ is a parameter capturing the concentration of the points in each cluster, $\sigma_{\mathrm{max}}$ and $\sigma_{\mathrm{min}}$ are the maximum and minimum standard deviation of the clusters around their means, and $\ell$ is the number of distinct mixing weights in the underlying clustering (after rounding them to the nearest power of $2$).
no code implementations • ICML 2020 • Sai Ganesh Nagarajan, David Balduzzi, Georgios Piliouras
Even simple games learning dynamics can yield chaotic behavior.