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Maximum Likelihood Estimation for Learning Populations of Parameters
Ramya Korlakai Vinayak · Weihao Kong · Gregory Valiant · Sham Kakade

Tue Jun 06:30 PM -- 09:00 PM PDT @ Pacific Ballroom #189
Consider a setting with $N$ independent individuals, each with an unknown parameter, $p_i \in [0, 1]$ drawn from some unknown distribution $P^\star$. After observing the outcomes of $t$ independent Bernoulli trials, i.e., $X_i \sim \text{Binomial}(t, p_i)$ per individual, our objective is to accurately estimate $P^\star$ in the sparse regime, namely when $t \ll N$. This problem arises in numerous domains, including the social sciences, psychology, health-care, and biology, where the size of the population under study is usually large yet the number of observations per individual is often limited. Our main result shows that, in this sparse regime where $t \ll N$, the maximum likelihood estimator (MLE) is both statistically minimax optimal and efficiently computable. Precisely, for sufficiently large $N$, the MLE achieves the information theoretic optimal error bound of $\mathcal{O}(\frac{1}{t})$ for $t < c\log{N}$, with regards to the earth mover's distance (between the estimated and true distributions). More generally, in an exponentially large interval of $t$ beyond $c \log{N}$, the MLE achieves the minimax error bound of $\mathcal{O}(\frac{1}{\sqrt{t\log N}})$. In contrast, regardless of how large $N$ is, the naive "plug-in" estimator for this problem only achieves the sub-optimal error of $\Theta(\frac{1}{\sqrt{t}})$. Empirically, we also demonstrate the MLE performs well on both synthetic as well as real datasets.

Author Information

Ramya Korlakai Vinayak (University of Washington)
Weihao Kong (Stanford University)
Gregory Valiant (Stanford University)
Sham Kakade (University of Washington)

Sham Kakade is a Washington Research Foundation Data Science Chair, with a joint appointment in the Department of Computer Science and the Department of Statistics at the University of Washington, and is a co-director for the Algorithmic Foundations of Data Science Institute. He works on the mathematical foundations of machine learning and AI. Sham's thesis helped in laying the foundations of the PAC-MDP framework for reinforcement learning. With his collaborators, his additional contributions include: one of the first provably efficient policy search methods, Conservative Policy Iteration, for reinforcement learning; developing the mathematical foundations for the widely used linear bandit models and the Gaussian process bandit models; the tensor and spectral methodologies for provable estimation of latent variable models (applicable to mixture of Gaussians, HMMs, and LDA); the first sharp analysis of the perturbed gradient descent algorithm, along with the design and analysis of numerous other convex and non-convex algorithms. He is the recipient of the IBM Goldberg best paper award (in 2007) for contributions to fast nearest neighbor search and the best paper, INFORMS Revenue Management and Pricing Section Prize (2014). He has been program chair for COLT 2011. Sham was an undergraduate at Caltech, where he studied physics and worked under the guidance of John Preskill in quantum computing. He then completed his Ph.D. in computational neuroscience at the Gatsby Unit at University College London, under the supervision of Peter Dayan. He was a postdoc at the Dept. of Computer Science, University of Pennsylvania , where he broadened his studies to include computational game theory and economics from the guidance of Michael Kearns. Sham has been a Principal Research Scientist at Microsoft Research, New England, an associate professor at the Department of Statistics, Wharton, UPenn, and an assistant professor at the Toyota Technological Institute at Chicago.

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