Give an example of a function that is
1. Lipschitz but not smooth
2. Smooth but not Lipschitz
3. Convex but neither smooth nor Lipschitz
From Understanding Machine Learning: From Theory to Algorithms,
1. 3.5: 2, 3
2. 4.5: 2
3. 12.6: 2, 3
From Convex Optimization
1. 2.1
2. 3.1

DS603: Robust Machine Learning

Elective, IIT Bombay, C-MInDS, 2025

Course Title: Robust Machine Learning
Instructor: Arjun Bhagoji
TA: Mohamad Hassan N C (mohamad.hassan@iitb.ac.in)
Time: Wednesday, Friday 9.30-11.00am
Room: LT103
Office Hours: 4.30-5.30pm on Wednesdays, CC120

DS606: Advances in Safety-Critical Machine Learning

Elective, IIT Bombay, C-MInDS, 2026

Course Title: Advances in Safety-Critical Machine Learning
Instructor: Arjun Bhagoji
TA: Anurag Deshpande, Prarabdh Shukla
Time: Wednesday, Friday 11.00-12.30am
Room: LT102
Office Hours: Wednesday 2-2.30pm

DS703: Introduction to Statistical Inference and Machine Learning

Core, IIT Bombay, C-MInDS, 2026

Course Title: Introduction to Statistical Inference and Machine Learning
Instructors: Parthe Pandit, Arjun Bhagoji
TA: Sreeranjini T M, Apurva Dhingra
Time: Monday, Tuesday, Thursday (11.35-12.30am)
Room: LT206
Office Hours: Arjun (2.30-3.30pm on Mondays in CC120/online), Parthe (4-5pm on Mondays in KR-320)

Practice Problems for DS703

Elective, IIT Bombay, C-MInDS, 2026

Probability

If $A$ and $B$ are events, and $X$ and $Y$ are the indicator random variables refined with respect to them. What is the minimal $\sigma$-algebra for the random variable $X+Y$.
Argue that $\mathbb{E} \exp(jX)$ is well-defined for all random variables $X$, where $j=\sqrt{-1}$ the imaginary number.
Find an example of random variables X, Y, Z taking values in {0,1} such that they are pairwise independent but not jointly independent, ie, P(X=1,Y=1,Z=1) ≠ P(X=1) P(Y=1) P(Z=1).
For general measures $\mu$ and an $A\in\mathcal{F}$, we say “$A\, \mu$-a.e.” to mean that $\mu(A^c) = 0$. If the measure mu is unambiguous, we just say “A a.e.” (a.e. is short for almost everywhere) example: $f\geq g$ a.e. means $\mu((f-g)^{-1}((-\infty,0)))=0$. similarly we say $f=g$ a.e. if $f\geq g$ a.e. and $f\leq g$ a.e. For probability measures, we say a.s. (almost surely) instead of a.e. Find an example of a sequence of non-negtive random variables for which $lim_{n\to\infty} P(X_n > 0) =0$, but $\lim \sup X_n = 1$ a.s.
For a non-negative random variable, show that $\mathbb{E}X = \int_{\mathbb{R}} P(X>t)\mathrm{d}t$. [Hint: First try showing it assuming that $P$ is absolutely continuous, i.e., it has a density function $f_X$.]
From One Thousand Exercises in Probability:
- Chapter 1: 1.2.2, 1.2.4, 1.3.1, 1.4.1, 1.4.3, 1.4.5, 1.5.1, 1.5.4, 1.5.5, 1.8.3, 1.8.4, 1.8.8, 1.8.9
- Chapter 2: 2.1.1, 2.1.2, 2.1.4, 2.1.5, 2.7.3
- Chapter 3: 3.2.1, 3.3.1, 3.3.6, 3.7.1, 3.7.4, 3.7.6
- Chapter 4: 4.5.8
- Chapter 7: 7.2.6, 7.5.1, 7.9.4, 7.11.2, 7.11.3, 7.11.8, 7.11.25