Search Talks

We will discuss $\rho$-equivariant conformal minimal immersions of finite energy into $\mathbb{CH}^2$. We prove that the induced metric is complete at a puncture $p$ if and only if the holonomy of $\rho$ around $p$ is parabolic. In this case $f$ is proper and $\partial f$ has pole at each puncture, so on the compactification $\Sigma$ we obtain a parabolic $\mathrm{PU}(2,1)$–Higgs bundle $(E,\Phi)$ with nilpotent residues and zero parabolic weights. This yields an Osserman-type extension principle for minimal surfaces in $\mathbb{CH}^2$ with complete ends. We further provide an algebro–geometric characterization of these immersions in terms of their associated parabolic Higgs data. This is a joint work with Indranil Biswas and John Loftin.

The unit sphere minimizes the first positive Laplacian eigenvalue among all compact n-dimensional manifolds with Ricci curvature bounded below by n−1, as a consequence of Lichnerowicz's Theorem. This naturally raises a question about the higher Laplacian eigenvalues: Does the unit sphere minimize all the Laplacian eigenvalues? In this talk, we will explore a class of manifolds whose Laplacian eigenvalues are strictly greater than the corresponding eigenvalues of the unit sphere. These examples provide evidence for such a general minimization in dimensions two and three.

Using a special Euler–Ramanujan identity and Wick rotation, we reveal how Scherk-type zero mean curvature surfaces in Lorentz–Minkowski space can be decomposed into superpositions of dilated helicoids and hyperbolic helicoids. These decompositions also extend to maximal codimension-2 surfaces, linking them to weakly untrapped and ∗-surfaces.

We consider the problem of finding the optimal value of n in the n-step temporal difference (TD) learning algorithm. Our objective function for the optimization problem is the average root mean squared error (RMSE). We find the optimal n by resorting to a model-free optimization technique involving a one-simulation simultaneous perturbation stochastic approximation (SPSA) based procedure. Whereas SPSA is a zeroth-order continuous optimization procedure, we adapt it to the discrete optimization setting by using a random projection operator. We prove the asymptotic convergence of the recursion by showing that the sequence of n-updates obtained using zeroth-order stochastic gradient search converges almost surely to an internally chain transitive invariant set of an associated differential inclusion. This results in convergence of the discrete parameter sequence to the optimal n in n-step TD. Through experiments, we show that the optimal value of n is achieved with our algorithm for arbitrary initial values. We further show using numerical evaluations that the proposed algorithm outperforms a well known discrete parameter stochastic optimization algorithm ‘Optimal Computing Budget Allocation (OCBA)’ on benchmark RL tasks. 

When a neural network becomes extremely wide or deep, its learning dynamics simplify and can be described by the same “mean-field” ideas that explain magnetism and fluids. I will walk through these ideas step-by-step, showing how they suggest practical recipes for initialization and optimization that scale smoothly from small models to cutting-edge transformers. I will also discuss neural scaling laws—empirical power-law rules that relate model size, data, and compute—and illustrate them with solvable toy models.

 In this work, we address the challenge of identifying the optimal arm in a stochastic multi-armed bandit scenario with the minimum number of arm pulls, given a predefined error probability in a fixed confidence setting. Our focus is on examining the asymptotic behavior of sample complexity and the distribution of arm weights upon termination, as the error threshold is scaled to zero, under confidence-interval based algorithms. Specifically, we analyze the asymptotic sample complexity and termination weight fractions for the well-known LUCB algorithm, and introduce a new variant, the LUCB Greedy algorithm. We demonstrate that the upper bounds on the sample complexities for both algorithms are asymptotically within a (universal) constant factor of the established lower bounds.

n causal inference, true causal order and the graph of causal interaction can be uniquely determined if you have sufficient interventional data. Interventions are local (randomized control trials) RCTs done where different variables, taken few or one at a time, in a causal graph are randomized. We consider a harder problem when the causal variables are not directly observed and are "latent". Instead, we observe a high dimensional transformation (as images etc.) of the true causal variables. Central problem in causal representation learning is to invert the unknown transformation between true causal variables and the observations up to coordinate wise scaling and permutation. We show that this is possible with enough interventional diversity by exploiting two key ideas: a) Represent interventional distributions in terms of their scores (gradient of likelihoods). b) The encoder-decoder pair that minimizes reconstruction loss and sparsifies the score difference in the latent space is the optimal pair. We show various versions of these results for linear transforms and general transforms with mild regularity assumptions on the diversity of interventions. We also will discuss empirical results on some simple image datasets.

Joint work with Burak Varici (CMU), Emre Acarturk (RPI), Abhishek Kumar (Amazon, ex-GDM), Ali Tajer (RPI)

Bandit convex optimization is a powerful framework for sequential decision-making, but existing algorithms with optimal regret guarantees are often too computationally expensive for high-dimensional problems. This talk introduces a simple and practical BCO algorithm, the Bandit Newton Step, which leverages second-order information for decision-making. We will show that our algorithm obtains an optimal $O(T^{1/2})$ regret bound for a large and practical class of functions that satisfy a condition we call “$\kappa$-convexity,” which includes linear, quadratic, and generalized linear losses. In addition to optimal regret, this method is the most efficient known algorithm for several well-studied applications including bandit logistic regression.

Furthermore, we'll discuss the extension of our method to online convex optimization with memory. We show that for loss functions with a certain affine structure, the extended algorithm attains optimal regret. This leads to an optimal regret algorithm for the bandit Linear-Quadratic (LQ) control problem under a fully adversarial noise model, resolving a key open question in the field. Finally, we contrast this result by proving that bandit problems with more general memory structures are fundamentally harder, establishing a tight $\Omega(T^{2/3})$ lower bound on regret.

The success of modern AI models defies classical theoretical wisdom. Classical theory recommended the use of convex optimization, and yet AI models learn by optimizing highly non-convex function. Classical theory prescribed to control model complexity and yet AI models are very complex, so complex that they often memorize the training data. Classical wisdom recommends a careful and interpretable choice of model architecture, and yet modern architectures rarely offer a parsimonious representation of a target distribution class.

The discovery that learning can take place in completely unexpected scenario poses beautiful conceptual challenges. I will try to survey recent work towards addressing them.