We consider the problem of score function estimation via empirical risk minimization. We first start with the question of inferring the score function of a probability measure $μ$ with density on the flat torus from a sample of distribution $μ$. We show that constraining the hypothesis space to a Sobolev ball is sufficient to prevent overfitting and obtaining minimax estimation rates. We then consider the problem of score function estimation in the context of score-based generative modeling. Again, under a conjecture tying the score estimation rates to the quality of the output of a score-based generative model, we obtain minimax rates for such an approach using score function estimators obtained by constraining the hypothesis class to a Sobolev ball.

We study online estimation for high-dimensional generalized linear models with streaming data. First, for the non-distributed setting, we propose a gradient-enhanced surrogate loss that approximates the cumulative loss using only historical summaries, which modifies and improves upon the existing renewable estimation approach for the same model in the high-dimensional setting, and removes the batch-number constraint in previous studies. We then extend the method to distributed streaming data under the master-client architecture, where batches are partitioned across sites and only summaries (gradient vectors) are exchanged. Instead of directing applying the popular method of Jordan et al. (2019) to the surrogate quadratic loss, our adjusted approach does not require the clients to compute the full surrogate loss. We derive non-asymptotic error bounds under the high-dimensional scaling, without the stringent constraint on the number of batches in the previous studies. Simulation results under linear and logistic models, together with a real-data application, show improved accuracy over existing renewable estimators.

We study a dynamic assortment problem on a two-sided service platform with incomplete information and heterogeneous customers in a discrete-time setting. In each period, a customer arrives seeking service, and the platform chooses an assortment of sellers to display. The customer then proposes a transaction to at most one seller in the assortment according to a multinomial logit choice model. After a fixed number of periods, sellers review the proposals they have received and each chooses at most one customer according to another multinomial logit choice model, after which the cycle repeats. A key challenge is that the platform does not know the choice-model parameters of either customers or sellers in advance. To our knowledge, this is the first study of a dynamic assortment problem in which both sides' choice parameters are unknown. We develop a data-driven algorithm that learns these parameters while optimizing the platform's objective over time. We evaluate performance using regret, which measures revenue loss relative to a clairvoyant benchmark that knows all parameters and customer arrivals in advance. We show that the algorithm's worst-case regret grows polylogarithmically over time, and we derive a matching lower bound, establishing its rate optimality.

There exists constantsC1,C2,C3,C4 depending onψ such that the following holds true.

Continual Learning (CL) poses a significant challenge in Artificial Intelligence, aiming to mirror the human ability to incrementally acquire knowledge and skills. While extensive research has focused on CL within the context of classification tasks, the advent of increasingly powerful generative models necessitates the exploration of Continual Learning of Generative models (CLoG). This paper advocates for shifting the research focus from classification-based CL to CLoG. We systematically identify the unique challenges presented by CLoG compared to traditional classification-based CL. We adapt three types of existing CL methodologies, replay-based, regularization-based, and parameter-isolation-based methods to generative tasks and introduce comprehensive benchmarks for CLoG that feature great diversity and broad task coverage. Our benchmarks and results yield intriguing insights that can be valuable for developing future CLoG methods. Additionally, we will release a codebase designed to facilitate easy benchmarking and experimentation in CLoG publicly at https://github.com/linhaowei1/CLoG. We believe that shifting the research focus to CLoG will benefit the continual learning community and illuminate the path for next-generation AI-generated content (AIGC) in a lifelong learning paradigm.

We consider the problem of estimating the mean of a random vector based on $N$ independent, identically distributed observations. We prove the existence of an estimator that has a near-optimal error in all directions in which the variance of the one dimensional marginal of the random vector is not too small: with probability $1-\delta$, the procedure returns $\wh{\mu}_N$ which satisfies that for every direction $u \in S^{d-1}$, \[ \inr{\wh{\mu}_N - \mu, u}\le \frac{C}{\sqrt{N}} \left( \sigma(u)\sqrt{\log(1/\delta)} + \left(\E\|X-\EXP X\|_2^2\right)^{1/2} \right)~, \] where $\sigma^2(u) = \var(\inr{X,u})$ and $C$ is a constant. To achieve this, we require only slightly more than the existence of the covariance matrix, in the form of a certain moment-equivalence assumption. The proof relies on novel bounds for the ratio of empirical and true probabilities that hold uniformly over certain classes of random variables.

When my children were young, we had a toy that they assembled consisting of seventy-five plastic interconnecting tunnel pieces, including having numerous tall ramps and winding paths, and when a marble was dropped into the topmost funnel it would be of great delight to all as we watched the marble roll throughout the structure. It was advertised via a slogan that said down the tube it goes, where the marble stops, nobody knows, and presumably helped teach my children about physics (well, it was actually mainly just a lot of fun). Being quite rambunctious, the kids sought out new ways to test the capabilities and limits of the toy. Putting one marble down the shoot was fun. Perhaps putting two marbles would be twice the fun! They tried this and it made them squeal with delight. If two marbles are twice the fun, certainly four marbles would quadruple the fun. They kept increasing the number of marbles and with each such increment the plastic contraption would shake and shimmy more so. How many marbles would the system withstand?

We consider the problem of bandit optimization, inspired by stochastic optimization and online learning problems with bandit feedback. In this problem, the objective is to minimize a global loss function of all the actions, not necessarily a cumulative loss. This framework allows us to study a very general class of problems, with applications in statistics, machine learning, and other fields. To solve this problem, we analyze the Upper-Confidence Frank-Wolfe algorithm, inspired by techniques for bandits and convex optimization. We give theoretical guarantees for the performance of this algorithm over various classes of functions, and discuss the optimality of these results.

Cécile B. Evans, How happy a Thing Can Be, 2014. The word'automation' is appearing in places that would have seemed unlikely to most people less than a decade ago: journalism, art, design or law. Robots and algorithms are being increasingly convincing at doing things just like humans. The Promise of Total Automation, an exhibition recently opened at Kunsthalle Wien in Vienna, looks at our troubled relationship with machines. Technical devices that were originally designed to serve and assist us and are now getting smarter and harder to control and comprehend.