We prove novel algorithmic guarantees for several online problems in the smoothed
analysis model. In this model, at each time step an adversary chooses an input distribution …

We consider the problem of dividing limited resources to individuals arriving over T rounds.
Each round has a random number of individuals arrive, and individuals can be …

The binary (or Ising) perceptron is a toy model of a single-layer neural network and can be
viewed as a random constraint satisfaction problem with a high degree of connectivity. The …

We study discrepancy minimization for vectors in ℝ n under various settings. The main result
is the analysis of a new simple random process in high dimensions through a comparison …

Random reshuffling, which randomly permutes the dataset each epoch, is widely adopted in
model training because it yields faster convergence than with-replacement sampling …

A set of T indivisible goods has to be allocated to a set of n agents with additive utilities, in a
way that is fair and efficient. A standard fairness concept is envy-freeness, which requires …

We give a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given
symmetric d× d matrices A 1,…, A n each with|| A i|| op≤ 1 and rank at most n/log3 n, one …

We study trade-offs between fairness and efficiency when allocating indivisible items online.
We attempt to minimize envy, the extent to which any agent prefers another's allocation to …

For many computational problems involving randomness, intricate geometric features of the
solution space have been used to rigorously rule out powerful classes of algorithms. This is …

Discrepancy theory has provided powerful tools for producing higher-quality objects which
“beat the union bound” in fundamental settings throughout combinatorics and computer …