I am a PhD candidate in the Statistics department at UCLA and a
member of the Statistical Machine
Learning Lab led by Quanquan
Gu. I'm interested in optimization and generalization in deep
learning and other problems in statistical learning. I also develop the deep learning algorithms
that power the Chatterbaby app.

**Recent news**

__2020__
* I've been awarded a

Dissertation
Year Fellowship by UCLA's Graduate Division.

* New

paper on agnostic PAC learning of a single neuron using gradient
descent is now on arXiv.

* New

paper accepted at

*Brain Structure and Function* from work with researchers at UCLA School of Medicine.

* I'll be (remotely) working at Amazon's

Alexa
AI group for the summer as a research intern, working on natural
language understanding.

* I'm reviewing for

NeurIPS 2020.

* I'm reviewing for

ICML 2020.

*2019*
* My paper with Yuan Cao and Quanquan Gu, "Algorithm-dependent
Generalization Bounds for Overparameterized Deep Residual Networks", was
accepted at NeurIPS 2019 (

arXiv
version,

NeurIPS
version).

I am currently a PhD candidate in the Statistics department at UCLA
and a member of the Statistical
Machine Learning Lab. I am supervised by Ying
Nian Wu from the Department of Statistics and Quanquan
Gu from the Department of Computer Science. I completed my
masters in mathematics at the University of British Columbia,
Vancouver, in May 2015. I was a member of the Probability
Group, and Ed
Perkins was my supervisor. Before that, I completed my
undergraduate degree in mathematics at McGill University in 2013.

You may find more information about me on my CV
(last updated May 2020).

Spring 2020: Stats 100C, Linear Models with Arash Amini.

Past teaching positions:

Fall 2019: Stats 102C, Monte Carlo Methods with Qing Zhou.

Summer 2016, Session C: Stats 10, Intro Statistics with Juana Sanchez.

Summer 2016, Session A: Stats 10, Intro Statistics with Miles Chen.

Fall 2016: Stats 100A, Introduction to Probability Theory with Ying
Nian Wu.

Winter 2016: Stats 100B, Introduction to Mathematical Statistics with
Jessica Li.

Since Fall 2017, I have been working as a graduate student researcher
(GSR) for Ariana Anderson at the Semel Institute for Neuroscience and
Human Behavior. For the 2016-2017 school year, I worked as a GSR for
Ariana Anderson and Monika Mellem.

#### Preprints

1.

**S. Frei**, Y. Cao, and Q. Gu.

Agnostic
learning of a single neuron with gradient descent. Preprint, 2020.

#### Refereed Conference Publications

2.

**S. Frei**, Y. Cao, and Q. Gu.

Algorithm-dependent
generalization bounds for overparameterized deep residual networks.
Conference on Neural Information Processing Systems (NeurIPS), 2019.

#### Journal Publications

3. A.E. Anderson, M. Diaz-Santos,

**S. Frei** *et al.*
Hemodynamic latency is associated with reduced intelligence across the
lifespan: an fMRI DCM study of aging, cerebrovascular integrity, and
cognitive ability. Brain Structure and Function, 2020

4.

**S. Frei** and E. Perkins.

A
lower bound for the critical probability in range-$R$
bond percolation on $\mathbb Z^d$ via SIR
epidemic methods. Electronic Journal of Probability 21(56), 2016.

5.

**S. Frei**, K. Lockwood, G. Stewart, J. Boyer, and B.S. Tilley.

On
thermal resistance in concentric residential geothermal heat
exchangers. Journal of Engineering Mathematics 86(1), 2014.

Below is a list of some of the projects I have worked on. Note
that this page utilizes MathJax
rendering of LaTeX code.

### SIR epidemics and range-R percolation on \(\mathbb{Z}^d\)

One can construct a discrete-time mathematical SIR (susceptible,
infected, recovered) epidemic on a graph \(G =
(V,E)\) by partitioning the elements in \(V\)
into one of three sites: susceptible \((x\in
\xi_n)\), infected \((x\in \eta_n)\),
and recovered \((x\in \rho_n)\). We start
from an initial configuration of susceptible and infected sites (say,
\(\eta_0 = \{ 0\}\), \(\xi_0
= V \setminus \eta_0\), and \(\rho_0=\emptyset\)),
and we consider a fixed parameter \(p\in (0,1)\)
that is the probability that any infected site \(x\in
\eta_n\) infects a neighboring susceptible site \(y\in
\xi_n\) (so that \((x,y)\in E)\).
Infected sites remain infected for one unit of time, and then become
`recovered’, and may no longer infect any other sites (or become
infected again); that is, \(x\in \eta_n\)
means \(x\in \rho_{n+1}\). We then let the
process run for times \(n=1, 2, \dots\) and
examine the behavior of the infected sites \(\eta_n\).

There are two possible behaviors for the epidemic: either the
epidemic dies out in finite time almost surely (i.e., \(\eta_n
= \emptyset\) eventually), or with positive probability the
epidemic continues forever, so that at any given time \(m\),
there is at least one site \(x\in \eta_m\).
In the latter case, we say that infection occurs in the epidemic.
Since each edge \((x,y)\) in the epidemic
process is `used’ precisely once—when either \(x\)
attempts to infect \(y\) or
vice-versa—there is a correspondence between infections occurring in a
graph \(G\) and percolation occurring in
the same graph: an infection occurs in \(G\)
iff percolation occurs in \(G\). Standard
percolation theory shows that there exists a critical value \(p_c(G)\)
for the probability \(p\) such that when \(p>p_c\), infection occurs, and for \(p<p_c\),
the epidemic dies out almost surely.

I studied this process on the graph \(\mathbb
Z^d\), with an edge set \(E = E_R\)
parameterized by a quantity \(R\), such
that there are edges between two sites \(x,y\in
\mathbb Z^d\) iff \(|x-y|_{\infty} \leq
R\), where \(|\cdot|_\infty\)
denotes distance in the \(\ell^\infty\)
metric. As the graph is parameterized by \(R\),
the corresponding critical value \(p_c\) is
likewise paramaterized by \(R\). In my
masters thesis, I proved a lower bound on the critical value for
percolation, as a function of \(R\), in
dimensions \(d=2\) and \(d=3\)
by studying the corresponding SIR epidemic. A paper on this result, in
collaboration with Ed Perkins, is available at the Electronic
Journal of Probability.

### Mathematical model of geothermal heating systems

Geothermal heat pump systems (also known as ground-source heat pump
systems, or GSHP) are an incredibly energy-efficient way to heat and
cool a home. Traditional heat
pumps rely on a heat exchanger that exchanges heat with air,
which functions as a heat source/sink. Geothermal heat pump
systems utilize the fact that just a few meters below the surface, the
ground is nearly constant temperature year-round, and hence use the
soil as a heat source/sink. The heat exchanger takes the form of
a liquid running through pipes that are burrowed underground.
As the temperature of the soil is near-constant year round, the heat
pump is also able to function effectively year-round, as the liquid
through the pipes deposits heat into the soil in the summer, and
receives energy from the soil in the winter. The pipes may be situated
in a number of networks, most of which are described at the Department
of Energy website.

The coefficient-of-performance, a common performance metric for
heating systems that is defined as the energy output (in heat) of a
system per energy input (in electricity to run the system), for GSHP
is approximately 4-7, while traditional heat pump systems that utilize
air as the heat exchange medium have coefficients of performance of
approximately 2. The main deterrent to widespread adoption of
ground source heat-pump systems is the large initial investment cost
(on the order of $10,000s) which is recuperated in energy savings over
approximately 10 years. The initial investment is largely due to
the drilling of deep (100-300 feet) boreholes that are approximately
1-2 feet in diameter that house the pipes that allow the water to
exchange heat with the soil through the highly conductive material
constituting the pipe walls. And, although it is well-known that
these systems are highly effective in certain
geological/climatological conditions, there is little knowledge of the
mathematics underpinning the process of heat exchange through the pipe
walls. It is particularly important to understand how deep one
must drill the boreholes in order for the heat-pump system to function
properly, as this is the main contribution to the installment cost.

I worked with Burt Tilley,
Kathryn Lockwood, Greg Stewart, and Justin Boyer to develop a
mathematical model of the heat exchange occurring between the water in
the pipes, the pipe walls, and the soil in a specific piping system
known as the concentric vertical pipe. In this pipe, water flows
down from the home through an inner cylindrical cavity, and flows back
up the pipe to the home through an outer cylindrical cavity. The
diagram to the left depicts the system in action during summer, as the
water decreases in temperature as it exchanges heat from the
soil. The pipes are designed so that the inner pipe material in
the region \(R_0^* < r^* < R_1^*\) has low conductivity while
the outer pipe material at \(r^* = R_2^*\) is highly conductive.
Our model studied the heat exchange as a function of design parameters
such as the spacing of the inner and outer pipes, the flow rate, and
the length of the pipe. We developed a characteristic length at
which heat exchange occurs in the soil, and found that one could drill
less deeply while still retaining the vast majority of heat exchange
ability of the system. These results
were published in the Journal of Engineering Mathematics in 2014.