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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 quantum minimum-finding routine from D\"urr 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 routine from Chen and de Wolf (ICALP'23). As one of our main contributions, we construct a quantum unitary to approximately compute the joining times to be searched over by the approximate quantum minimum finding. Since the joining times are no longer exactly computed, it is no longer clear that the resulting approximate quantum algorithm obtains a good solution. As our second main contribution, we prove, via an approximate version of the KKT conditions and a duality gap, that the LARS algorithm (and thus 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 approximately computed. Moreover, 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 $d$. Finally, we propose a dequantised algorithm that also retains the polylogarithmic dependence on $n$, albeit with the linear scaling on $d$ from the standard LARS algorithm., Comment: 48 pages. v2: several improvements, typos fixed, references added, fixed a bug in Theorem 28, exponentially improved the complexity dependence on the number of observations $n$ for a random Gaussian input matrix

Adapters are widely popular parameter-efficient transfer learning approaches in natural language processing that insert trainable modules in between layers of a pre-trained language model. Apart from several heuristics, however, there has been a lack of studies analyzing the optimal number of adapter parameters needed for downstream applications. In this paper, we propose an adapter pruning approach by studying the tropical characteristics of trainable modules. We cast it as an optimization problem that aims to prune parameters from the adapter layers without changing the orientation of underlying tropical hypersurfaces. Our experiments on five NLP datasets show that tropical geometry tends to identify more relevant parameters to prune when compared with the magnitude-based baseline, while a combined approach works best across the tasks., Comment: Accepted at EMNLP 2023, Findings

We introduce algebraic machine reasoning, a new reasoning framework that is well-suited for abstract reasoning. Effectively, algebraic machine reasoning reduces the difficult process of novel problem-solving to routine algebraic computation. The fundamental algebraic objects of interest are the ideals of some suitably initialized polynomial ring. We shall explain how solving Raven's Progressive Matrices (RPMs) can be realized as computational problems in algebra, which combine various well-known algebraic subroutines that include: Computing the Gr\"obner basis of an ideal, checking for ideal containment, etc. Crucially, the additional algebraic structure satisfied by ideals allows for more operations on ideals beyond set-theoretic operations. Our algebraic machine reasoning framework is not only able to select the correct answer from a given answer set, but also able to generate the correct answer with only the question matrix given. Experiments on the I-RAVEN dataset yield an overall $93.2\%$ accuracy, which significantly outperforms the current state-of-the-art accuracy of $77.0\%$ and exceeds human performance at $84.4\%$ accuracy., Comment: Accepted at IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) 2023. 30 pages, 7 figures (including supplementary material). First three authors contributed equally. Code is available at: https://github.com/Xu-Jingyi/AlgebraicMR

We propose a new approach, Knowledge Distillation using Optimal Transport (KNOT), to distill the natural language semantic knowledge from multiple teacher networks to a student network. KNOT aims to train a (global) student model by learning to minimize the optimal transport cost of its assigned probability distribution over the labels to the weighted sum of probabilities predicted by the (local) teacher models, under the constraints, that the student model does not have access to teacher models' parameters or training data. To evaluate the quality of knowledge transfer, we introduce a new metric, Semantic Distance (SD), that measures semantic closeness between the predicted and ground truth label distributions. The proposed method shows improvements in the global model's SD performance over the baseline across three NLP tasks while performing on par with Entropy-based distillation on standard accuracy and F1 metrics. The implementation pertaining to this work is publicly available at: https://github.com/declare-lab/KNOT., Comment: COLING 2022

We showcase a topological mapping framework for a challenging indoor warehouse setting. At the most abstract level, the warehouse is represented as a Topological Graph where the nodes of the graph represent a particular warehouse topological construct (e.g. rackspace, corridor) and the edges denote the existence of a path between two neighbouring nodes or topologies. At the intermediate level, the map is represented as a Manhattan Graph where the nodes and edges are characterized by Manhattan properties and as a Pose Graph at the lower-most level of detail. The topological constructs are learned via a Deep Convolutional Network while the relational properties between topological instances are learnt via a Siamese-style Neural Network. In the paper, we show that maintaining abstractions such as Topological Graph and Manhattan Graph help in recovering an accurate Pose Graph starting from a highly erroneous and unoptimized Pose Graph. We show how this is achieved by embedding topological and Manhattan relations as well as Manhattan Graph aided loop closure relations as constraints in the backend Pose Graph optimization framework. The recovery of near ground-truth Pose Graph on real-world indoor warehouse scenes vindicate the efficacy of the proposed framework., Comment: Accepted to ICRA 2020. Project Page: https://github.com/Shubodh/ICRA2020

We propose new Degroot-type social learning models with feedback in a continuous time, to investigate the effect of a noisy information source on consensus formation in a social network. Unlike the standard Degroot framework, noisy information models destroy consensus formation. On the other hand, the noisy opinion dynamics converge to the equilibrium distribution that encapsulates correlations among agents' opinions. Interestingly, such an equilibrium distribution is also a non-equilibrium steady state (NESS) with a non-zero probabilistic current loop. Thus, noisy information source leads to a NESS at long times that encodes persistent correlated opinion dynamics of learning agents. Our model provides a simple realization of NESS in the context of social learning. Other phenomena such as synchronization of opinions when agents are subject to a common noise are also studied., Comment: 4 figures

We develop original models to study interacting agents in financial markets and in social networks. Within these models randomness is vital as a form of shock or news that decays with time. Agents learn from their observations and learning ability to interpret news or private information in time-varying networks. Under general assumption on the noise, a limit theorem is developed for the generalised DeGroot framework for certain type of conditions governing the learning. In this context, the agents beliefs (properly scaled) converge in distribution that is not necessarily normal. Fresh insights are gained not only from proposing a new setting for social learning models but also from using different techniques to study discrete time random linear dynamical systems., Comment: 34 pages, 8 figures

Black-Scholes (BS) is the standard mathematical model for option pricing in financial markets. Option prices are calculated using an analytical formula whose main inputs are strike (at which price to exercise) and volatility. The BS framework assumes that volatility remains constant across all strikes, however, in practice it varies. How do traders come to learn these parameters? We introduce natural models of learning agents, in which they update their beliefs about the true implied volatility based on the opinions of other traders. We prove convergence of these opinion dynamics using techniques from control theory and leader-follower models, thus providing a resolution between theory and market practices. We allow for two different models, one with feedback and one with an unknown leader., Comment: 19 pages, 3 figures