Research

I develop algorithms and mathematical foundations for machine learning, optimization, and control. I co-developed AdaGrad, which helped establish adaptive gradient methods in machine learning. A growing focus is evaluating and aligning AI systems that may surpass their human supervisors.

AI evaluation and alignment

At Princeton AI Alignment and Safety (AIAS), we use game theory, mechanism design, and online learning to evaluate capabilities beyond human expertise and design incentives for reliable behavior.

• Measuring Intelligence Beyond Human Scale (SepaRank): model-generated challenges for relative evaluation without human-authored benchmarks or a semantic judge.
• AI Alignment via Incentives and Correction: adapting rewards for solver–auditor interactions using bandit feedback.

Online learning and optimization

I study efficient algorithms for sequential decision-making, focusing on regret minimization, adaptive optimization, and bandit feedback, with connections to game theory and reinforcement learning.

Book: Introduction to Online Convex Optimization.

Control and sequence modeling

We develop online nonstochastic control and spectral methods for learning dynamical systems and modeling long sequences.

• Universal Sequence Preconditioning: memory-efficient prediction using orthogonal polynomials, with a second-order extension.
• Universal Learning of Nonlinear Dynamics: provable prediction through observation-based spectral filtering.
• Spectral Transformers: neural sequence architectures; FutureFill: fast generation for convolutional sequence models.

Selected contributions

• AdaGrad and adaptive regularization
  AdaGrad adapts updates to the geometry of past gradients. This work introduced an adaptive regularization framework with regret guarantees. See this blog post for background.

• Online Newton Step and logarithmic regret
  Connecting optimization curvature to fast online learning rates, Online Newton Step achieves logarithmic regret for exp-concave losses. It provides an efficient online counterpart of Newton’s method and extends ideas from universal portfolio selection.

• Sublinear-time optimization for machine learning and semidefinite programming
  Using randomized sampling and online learning, these algorithms approximately solve problems including linear classification, minimum enclosing balls, and a class of semidefinite programs in less time than reading the full input.

• Sparse approximate solutions to semidefinite programs
  Addressing the computational cost of large-scale semidefinite optimization, this simple algorithm constructs low-rank approximate solutions with efficient representations.

• Linearly convergent projection-free optimization
  Frank-Wolfe methods replace expensive projections with linear optimization. This method achieves linear convergence for smooth, strongly convex objectives over polyhedral sets with one linear optimization step per iteration. Iteration counts grow logarithmically with inverse target accuracy and also depend on the objective and domain geometry.

• Online nonstochastic control
  Under adversarial disturbances and changing costs, this framework uses regret minimization to learn disturbance-based control policies, competing with a benchmark policy class in hindsight. See Introduction to Online Control for theory and algorithms.

See Google Scholar for all publications and our group webpage for research updates.