Fridays 3:30pm in KAP 414; tea is usually provided at 3:00pm

Organizer: Stanislav Minsker

For seminars/colloquia on other topics, see the Department of Mathematics webpage.

Older seminars: Spring 2017, Fall 2016,  Spring 2016,  Fall 2015,  Spring 2015,  Fall 2014,  2013-2014

  • Fall 2017 seminars

    August 25: Christian Keller (University of Michigan, Department of Mathematics)

    Path-dependent Hamilton-Jacobi equations in infinite dimensions

    Abstract: We propose notions of minimax and viscosity solutions for a class of fully nonlinear path-dependent PDEs with nonlinear, monotone, and coercive operators on Hilbert space. Our main result is well-posedness (existence, uniqueness, and stability) for minimax solutions. A particular novelty of our approach is a suitable combination of minimax and viscosity solution techniques. Thereby, we establish a comparison principle for path-dependent PDEs under conditions that are weaker even in the finite-dimensional case. In contrast to most of the related works on PDEs in infinite dimensions, perturbed optimization is entirely avoided. The path-dependent setting itself enables us to circumvent the lack of compactness in infinite-dimensional Hilbert spaces. As an application, our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of (delay) evolution equations in the variational framework. Furthermore, differential games associated to such evolution equations can be investigated following the Krasovskii-Subbotin approach similarly as in finite dimensions. Our results are also potentially useful for applications to large deviations. This talk is based on joint work with Erhan Bayraktar.

    September 1: Wuchen Li (UCLA, Department of Mathematics)

    Optimal transport on finite graphs

    Optimal transport theory provides powerful tools in both mathematics and applications. In this talk, we consider similar matters on finite graphs. Various recent developments related to Fokker-Planck equations, Shannon-Boltzmann entropy, Fisher information, as well as Wasserstein metric on graphs will be presented. Some applications are presented, including evolutionary games and “geometry” of finite graphs.

    September 8: Wen-Xin Zhou (UC San Diego, Department of Mathematics)

    Robust estimation and inference via multiplier bootstrap

    Massive data are often contaminated by outliers and heavy-tailed errors. In the presence of heavy-tailed data, finite sample properties of the least squares-based methods, typified by the sample mean, are suboptimal both theoretically and empirically. To address this challenge, we propose the adaptive Huber regression for robust estimation and inference. The key observation is that the robustification parameter should adapt to sample size, dimension and moments for optimal tradeoff between bias and robustness. For heavy-tailed data with bounded $(1+\delta)$-th moment for some $\delta>0$, we establish a sharp phase transition for robust estimation of regression parameters in both finite dimensional and high dimensional settings: when $\delta \geq 1$, the estimator achieves sub-Gaussian rate of convergence without sub-Gaussian assumptions, while only a slower rate is available in the regime $0<\delta <1$ and the transition is smooth and optimal.
    In addition, non-asymptotic Bahadur representation and Wilks’ expansion for finite sample inference are derived when higher moments exist. Based on these results, we investigate the theoretical underpinnings of both classical and contemporary statistical inference problems, where heavy-tailed data are present. In particular, we focus on uncertainty quantification methodologies, including the construction of confidence sets and large-scale simultaneous hypothesis testing. We demonstrate that the adaptive Huber regression, combined with the multiplier bootstrap procedure, provides a useful robust alternative to the method of least squares. The idea of adaptivity is to let data, which are probably collected with low quality and exhibit heavy tails, to influence the choice of method by which they are analyzed. Together, the theoretical and empirical results reveal the effectiveness of the proposed method, and highlight the importance of having statistical methods that are robust to violations of the assumptions underlying their use.

    September 15: Jason Schweinsberg (UC San Diego, Department of Mathematics)

    Rigorous results for a population model with selection

    We consider a model of a population of fixed size N in which each individual acquires beneficial mutations at a constant rate.  Each individual dies at rate one, and when a death occurs, an individual is chosen at random with probability proportional to the individual’s fitness to give birth.  We obtain rigorous results for the rate at which mutations accumulate in the population, the distribution of the fitness levels of individuals in the population at a given time, and the genealogy of the population. Our results confirm nonrigorous predictions of Desai and Fisher (2007), Desai, Walczak, and Fisher (2013), and Neher and Hallatschek (2013).

    September 22: Ilya Mironov (Google)

    Differential Privacy: From Principled Foundations to Your Browser

    We survey progress in understanding of privacy in statistical databases over the last 10+ years, starting with early negative results followed by emergence of the notion of differential privacy and its variants. In the second half of the talk we cover uses of differential privacy in the Chrome browser, and its recent applications in machine learning tasks such as text and image recognition.

    Bio: Ilya Mironov is a Staff Research Scientist in Google Brain. After completing his PhD at Stanford in 2003, he joined Microsoft Research Silicon Valley, where he worked on cryptography, cryptanalysis, and privacy until 2014.

    September 29: Jianfeng Zhang (USC, Department of Mathematics)

    A Martingale Approach for Fractional Brownian Motions and Related Path Dependent PDEs
     
    In this talk we consider dynamic backward problems in a framework where the forward state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such a problem is important in financial markets with rough volatility, as confirmed by some empirical studies. These processes are neither Markov processes nor semimartingales. Our main result is a functional Ito formula, extending the seminal work of Dupire to our more general framework. In particular, unlike in Dupire’s work where one needs only to consider the stopped paths, here we need to concatenate the observed path up to the current time with certain smooth curve derived from the distribution of the future paths. This new feature is due to some intrinsic time inconsistency involved in this framework. We then derive the path dependent PDEs for the backward problems. The talk is based on a joint work with Frederi Viens.

    October 6: Kenneth Alexander (USC, Department of Mathematics)

    Intersections of independent renewal processes

    We consider the intersection (that is, the common renewals) of two independent renewal processes, $\rho=\tau\cap\sigma$. This seemingly natural and classical problem, which has received relatively little attention in the literature, arises in the context of a pinning model in statistical mechanics. Assuming that $P(\tau_1 = n ) = \varphi(n)\, n^{-(1+\alpha)}$ and $P(\sigma_1 = n ) = \tilde\varphi(n)\, n^{-(1+ \tilde\alpha)}$ for some $\alpha,\tilde\alpha \geq 0$ and some slowly varying $\varphi,\tilde\varphi$, we find the asymptotic behavior of $P(\rho_1=n)$. The result may be viewed as a kind of inverse of standard renewal theorems, as we determine probabilities $P(\rho_1=n)$ while knowing asymptotically the renewal mass function $P(n\in\rho)=P(n\in\tau)P(n\in\sigma)$. The work is joint with Quentin Berger.

    October 13: Larry Goldstein (USC, Department of Mathematics)

    Non asymptotic distributional bounds for the Dickman approximation of the running time of the Quickselect algorithm

    Given a non-negative random variable $W$ and $\theta>0$, let the generalized Dickman transformation map the distribution of $W$ to that of $W^*=_d U^{1/\theta}(W+1)$, where $U \sim {\cal U}[0,1]$, a uniformly distributed variable on the unit interval, independent of $W$, and $=_d$ denotes equality in distribution. It is well known that $W^*$ and $W$ are equal in distribution if and only if $W$ has the generalized Dickman distribution ${\cal D}_\theta$. By use of Stein’s method, one can show that the Wasserstein distance $d_1$ between $W$, a non-negative random variable with finite mean, and $D_\theta$ having distribution ${\cal D}_\theta$ obeys the inequality

    $d_1(W,D_\theta) \le (1+\theta)d_1(W,W^*).$

    The specialization of this bound to the case $\theta=1$ and coupling constructions yield

    $d_1(W_n,D) \le \frac{8\log (en/2)+2}{n} \quad \mbox{for all $n \ge 1$, where} \quad W_n=\frac{1}{n}C_n-1$,

    with $C_n$ the number of comparisons made by the Quickselect algorithm to find the smallest element of a list of $n$ distinct numbers. The running time for Quickselect to locate the $m^{th}$ smallest element of the list obeys a similar bound, and together recover the results of Hwang and Tsai (2002) that show distributional convergence of $W_n$ to the standard Dickman distribution in the asymptotic regime $m=o(n)$.

    October 20: Meisam Razaviyayn (USC, Department of Industrial & Systems Engineering)

    Two non-convex optimization problems: learning deep models and tuning hyper-parameters

    In this talk, we study two important classes of optimization problems arising in machine learning applications: 

    1) Learning deep models and 2) Tuning hyper-parameters. In the first part of the talk, we consider the problem of training deep models. These models maps the input (feature) data to the output (response) vector through the composition of multiple simple mappings. Training feedforward multi-layer neural networks or low rank matrix decomposition are examples of such problems. We introduce a mathematical framework for analyzing the local optima of training these deep models. In particular, by leveraging the local openness property of the resulting mapping, we provide sufficient conditions under which  the local optima of the objective function are globally optimal. Our result leads to the characterization of local optima of linear feedforward network and provides sufficient conditions for the existence of no spurious local optima under hierarchical non-linear neural networks.
    In the second part of the talk, we consider the problem of optimizing the cross validation (CV) for a given learning problem.  We first develop a computationally efficient approximate for CV and provide theoretical guarantees for its performance. Then we use our approximate to provide an optimization algorithm for finding the optimal hyper-parameters in the empirical risk minimization framework. In our numerical experiments, we illustrate the accuracy and efficiency of our approximate as well as our proposed framework for the optimal regularizer.

    This is a joint work with Maher Nouiehed (USC), Ahmad Beirami (MIT),  Shahin Shahrampour (Harvard), Vahid Tarokh (Harvard).

    October 27: Sanjoy Dasgupta (UC San Diego, Computer Science and Engineering)

    Learning from partial correction

    We introduce a new model of interactive learning in which an expert examines the predictions of a learner and partially fixes them if they are wrong. Although this kind of feedback is not i.i.d., we show statistical generalization bounds on the quality of the learned model. This is joint work with Mike Luby.

    November 3: David X. Li (Shanghai Advanced Institute of Finance (SAIF) and Shanghai Jiaotong University (SJTU))

    Theoretical Problems in Credit Portfolio Modeling

    This talk first provides an overview about the copula function approach to credit portfolio modeling. Some of the key theoretical deficiency in its current framework is then highlighted. Finally, some initial result based on the equilibrium approach to the credit portfolio modeling is presented. This new approach is based on the extension of the copula function for random variables to the copula function for stochastic processes. The basic definition, properties of copulas for stochastic processes are discussed. This new approach allows us to theoretically link our credit portfolio modeling with our classical equity portfolio modeling under the CAPM setting.

    November 10: Vladimir V. Ulyanov (Higher School of Economics, Moscow)

    On confidence sets for projectors of a covariance matrix

    We offer a bootstrap procedure to get the sharp confidence sets for the projectors of a covariance matrix  from the given sample. This procedure could be applied for small or moderate sample size and large dimension of observations. The main result states the validity of the proposed procedure for finite samples with an explicit error bound of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high dimensions.
    These are the joint results with V.Spokoiny (WIAS, Berlin) and A.Naumov (Skoltech, Moscow).

    November 10: Alexey Naumov (Skoltech, Moscow)

    Big ball probability with applications in statistical inference 

    We derive the bounds on the Kolmogorov distance between probabilities of two Gaussian elements to hit a ball in a Hilbert space. The key property of these bounds is that they are dimensional-free and depend on the nuclear (Schatten-one) norm of the difference between the covariance operators of the elements. We are also interested in the anticoncentration bound for a squared norm of a non-centered Gaussian element in a Hilbert space. All bounds are sharp and cannot be improved in general. We provide a list of motivation examples and applications in statistical inference for the derived results as well. The talk is based on joint results with F. Götze (Bielefeld University), V. Spokoiny (WIAS) and V. Ulyanov(Moscow State University).

    December 1: Ujan Gangopadhyay (Department of Mathematics, USC)

    We consider an urn model where the replacement matrices are random. The replacement matrices need neither be independent, nor identically distributed. However, we assume that the replacement matrices are independent of the color drawn in the same round conditioned on the entire past. We also assume the matrices to have only first moment finite, unlike the usual second moment assumption in the literature. We further require the conditional expectation of the replacement matrix given the past to be close to an (not necessarily nonrandom) irreducible matrix in some appropriate sense. We do not require any of the matrices to be balanced.

    When the replacement matrices have $p>1$ finite moments, we prove almost sure convergence of the proportion vector, while the convergence is in probability when the replacement matrices have only first finite moment. We also consider the growth rates of composition vectors and count vectors. We use stochastic approximation to analyze the model. We develop a new version of stochastic approximation with random step size and driving function. The related differential equation is of Lotka-Volterra type and can be analyzed directly.

    This is a joint work with Krishanu Maulik.