Debbie Lim

Andris Ambainis, Hartmut Klauck, Debbie Lim

Similarly to the Chomsky hierarchy, we offer a classification of communication complexity measures such that these measures are organized into equivalence classes. Different from previous attempts of this endeavor, we consider two communication complexity measures as equivalent, if, when one is constant, then the other is constant as well, and vice versa. Most previous considerations of similar topics have been using polylogarithmic input length as a defining characteristic of equivalence. In this paper, two measures ${\cal C}_1, {\cal C}_2$ are constant-equivalent, if and only if for all total Boolean (families of) functions $f:\{0, 1\}^n\times\{0, 1\}^n\rightarrow \{0, 1\}$ we have ${\cal C}_1(f)=O(1)$ if and only if ${\cal C}_2(f)=O(1)$. We identify five equivalence classes according to the above equivalence relation. Interestingly, the classification is counter-intuitive in that powerful models of communication are grouped with weak ones, and seemingly weaker models end up on the top of the hierarchy.

Joao F. Doriguello, Debbie Lim, Chi Seng Pun et al.

We present a novel quantum high-dimensional linear regression algorithm with an $\ell_1$-penalty based on the classical LARS (Least Angle Regression) pathwise algorithm. Similarly to available classical algorithms for Lasso, our quantum algorithm provides the full regularisation path as the penalty term varies, but quadratically faster per iteration under specific conditions. A quadratic speedup on the number of features $d$ is possible by using the simple quantum minimum-finding subroutine from Dürr and Hoyer (arXiv'96) in order to obtain the joining time at each iteration. We then improve upon this simple quantum algorithm and obtain a quadratic speedup both in the number of features $d$ and the number of observations $n$ by using the approximate quantum minimum-finding subroutine from Chen and de Wolf (ICALP'23). In order to do so, we approximately compute the joining times to be searched over by the approximate quantum minimum-finding subroutine. As another main contribution, we prove, via an approximate version of the KKT conditions and a duality gap, that the LARS algorithm (and therefore our quantum algorithm) is robust to errors. This means that it still outputs a path that minimises the Lasso cost function up to a small error if the joining times are only approximately computed. Furthermore, we show that, when the observations are sampled from a Gaussian distribution, our quantum algorithm's complexity only depends polylogarithmically on $n$, exponentially better than the classical LARS algorithm, while keeping the quadratic improvement on $d$. Moreover, we propose a dequantised version of our quantum algorithm that also retains the polylogarithmic dependence on $n$, albeit presenting the linear scaling on $d$ from the standard LARS algorithm. Finally, we prove query lower bounds for classical and quantum Lasso algorithms.

Andris Ambainis, Joao F. Doriguello, Debbie Lim

We propose novel classical and quantum online algorithms for learning finite-horizon and infinite-horizon average-reward Markov Decision Processes (MDPs). Our algorithms are based on a hybrid exploration-generative reinforcement learning (RL) model wherein the agent can, from time to time, freely interact with the environment in a generative sampling fashion, i.e., by having access to a "simulator". By employing known classical and new quantum algorithms for approximating optimal policies under a generative model within our learning algorithms, we show that it is possible to avoid several paradigms from RL like "optimism in the face of uncertainty" and "posterior sampling" and instead compute and use optimal policies directly, which yields better regret bounds compared to previous works. For finite-horizon MDPs, our quantum algorithms obtain regret bounds which only depend logarithmically on the number of time steps $T$, thus breaking the $O(\sqrt{T})$ classical barrier. This matches the time dependence of the prior quantum works of Ganguly et al. (arXiv'23) and Zhong et al. (ICML'24), but with improved dependence on other parameters like state space size $S$ and action space size $A$. For infinite-horizon MDPs, our classical and quantum bounds still maintain the $O(\sqrt{T})$ dependence but with better $S$ and $A$ factors. Nonetheless, we propose a novel measure of regret for infinite-horizon MDPs with respect to which our quantum algorithms have $\operatorname{poly}\log{T}$ regret, exponentially better compared to classical algorithms. Finally, we generalise all of our results to compact state spaces.

Debbie Lim, Yixian Qiu, Patrick Rebentrost et al.

Logistic regression, the Support Vector Machine (SVM), and least squares are well-studied methods in the statistical and computer science community, with various practical applications. High-dimensional data arriving on a real-time basis makes the design of online learning algorithms that produce sparse solutions essential. The seminal work of \hyperlink{cite.langford2009sparse}{Langford, Li, and Zhang (2009)} developed a method to obtain sparsity via truncated gradient descent, showing a near-optimal online regret bound. Based on this method, we develop a quantum sparse online learning algorithm for logistic regression, the SVM, and least squares. Given efficient quantum access to the inputs, we show that a quadratic speedup in the time complexity with respect to the dimension of the problem is achievable, while maintaining a regret of $O(1/\sqrt{T})$, where $T$ is the number of iterations.