Optimal transport (OT) theory describes general principles to define and select, among many possible choices, the most efficient way to map a probability measure onto another. That theory has been mostly used to estimate, given a pair of source and target probability measures $(\mu, \nu)$, a parameterized map $T_\theta$ that can efficiently map $\mu$ onto $\nu$. In many applications, such as predicting cell responses to treatments, pairs of input/output data measures $(\mu, \nu)$ that define optimal transport problems do not arise in isolation but are associated with a context $c$, as for instance a treatment when comparing populations of untreated and treated cells. To account for that context in OT estimation, we introduce CondOT, a multi-task approach to estimate a family of OT maps conditioned on a context variable, using several pairs of measures $\left(\mu_i, \nu_i\right)$ tagged with a context label $c_i$. CondOT learns a global map $\mathcal{T}_\theta$ conditioned on context that is not only expected to fit all labeled pairs in the dataset $\left\{\left(c_i,\left(\mu_i, \nu_i\right)\right)\right\}$, i.e., $\mathcal{T}_\theta\left(c_i\right) \sharp \mu_i \approx \nu_i$, but should also generalize to produce meaningful maps $\mathcal{T}_\theta\left(c_{\text {new }}\right)$ when conditioned on unseen contexts $c_{\text {new }}$. Our approach harnesses and provides a novel usage for partially input convex neural networks, for which we introduce a robust and efficient initialization strategy inspired by Gaussian approximations. We demonstrate the ability of CondOT to infer the effect of an arbitrary combination of genetic or therapeutic perturbations on single cells, using only observations of the effects of said perturbations separately.
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Meta-learning and few-shot prompting are viable methods to induce certain types of compositional behaviour. However, these methods can be very sensitive to the choice of support examples used. Choosing good supports from the training data for a given test query is already a difficult problem, but in some cases solving this may not even be enough. We consider a grounded language learning problem (gSCAN) where good support examples for certain test splits might not even exist in the training data, or would be infeasible to search for. We design an agent which instead generates possible supports which are relevant to the test query and current state of the world, then uses these supports via meta-learning to solve the test query. We show substantially improved performance on a previously unsolved compositional behaviour split without a loss of performance on other splits. Further experiments show that in this case, searching for relevant demonstrations even with an oracle function is not sufficient to attain good performance when using meta-learning.
Channel estimation and data transmission constitute the most fundamental functional modules of multiple-input multiple-output (MIMO) communication systems. The underlying key tasks corresponding to these modules are training sequence optimization and transceiver optimization. Hence, we jointly optimize the linear transmit precoder and the training sequence of MIMO systems using the metrics of their effective mutual information (MI), effective mean squared error (MSE), effective weighted MI, effective weighted MSE, as well as their effective generic Schur-convex and Schur-concave functions. Both statistical channel state information (CSI) and estimated CSI are considered at the transmitter in the joint optimization. A unified framework termed as joint matrix-monotonic optimization is proposed. Based on this, the optimal precoder matrix and training matrix structures can be derived for both CSI scenarios. Then, based on the optimal matrix structures, our linear transceivers and their training sequences can be jointly optimized. Compared to state-of-the-art benchmark algorithms, the proposed algorithms visualize the bold explicit relationships between the attainable system performance of our linear transceivers conceived and their training sequences, leading to implementation ready recipes. Finally, several numerical results are provided, which corroborate our theoretical results and demonstrate the compelling benefits of our proposed pilot-aided MIMO solutions.
Recent advances in machine learning-aided lossy compression are incorporating perceptual fidelity into the rate-distortion theory. In this paper, we study the rate-distortion-perception trade-off when the perceptual quality is measured by the total variation distance between the empirical and product distributions of the discrete memoryless source and its reconstruction. We consider the general setting, where two types of resources are available at both the encoder and decoder: a common side information sequence, correlated with the source sequence, and common randomness. We consider both the strong perceptual constraint and the weaker empirical perceptual constraint. The required communication rate for achieving the distortion and empirical perceptual constraint is the minimum conditional mutual information, and similar result holds for strong perceptual constraint when sufficient common randomness is provided and the output along with the side information is constraint to an independent and identically distributed sequence.
Algorithmic stability is an important notion that has proven powerful for deriving generalization bounds for practical algorithms. The last decade has witnessed an increasing number of stability bounds for different algorithms applied on different classes of loss functions. While these bounds have illuminated various properties of optimization algorithms, the analysis of each case typically required a different proof technique with significantly different mathematical tools. In this study, we make a novel connection between learning theory and applied probability and introduce a unified guideline for proving Wasserstein stability bounds for stochastic optimization algorithms. We illustrate our approach on stochastic gradient descent (SGD) and we obtain time-uniform stability bounds (i.e., the bound does not increase with the number of iterations) for strongly convex losses and non-convex losses with additive noise, where we recover similar results to the prior art or extend them to more general cases by using a single proof technique. Our approach is flexible and can be generalizable to other popular optimizers, as it mainly requires developing Lyapunov functions, which are often readily available in the literature. It also illustrates that ergodicity is an important component for obtaining time-uniform bounds -- which might not be achieved for convex or non-convex losses unless additional noise is injected to the iterates. Finally, we slightly stretch our analysis technique and prove time-uniform bounds for SGD under convex and non-convex losses (without additional additive noise), which, to our knowledge, is novel.
Sequential transfer optimization (STO), which aims to improve optimization performance by exploiting knowledge captured from previously-solved optimization tasks stored in a database, has been gaining increasing research attention in recent years. However, despite significant advancements in algorithm design, the test problems in STO are not well designed. Oftentimes, they are either randomly assembled by other benchmark functions that have identical optima or are generated from practical problems that exhibit limited variations. The relationships between the optimal solutions of source and target tasks in these problems are manually configured and thus monotonous, limiting their ability to represent the diverse relationships of real-world problems. Consequently, the promising results achieved by many algorithms on these problems are highly biased and difficult to be generalized to other problems. In light of this, we first introduce a few rudimentary concepts for characterizing STO problems (STOPs) and present an important problem feature overlooked in previous studies, namely similarity distribution, which quantitatively delineates the relationship between the optima of source and target tasks. Then, we propose general design guidelines and a problem generator with superior extendibility. Specifically, the similarity distribution of a problem can be systematically customized by modifying a parameterized density function, enabling a broad spectrum of representation for the diverse similarity relationships of real-world problems. Lastly, a benchmark suite with 12 individual STOPs is developed using the proposed generator, which can serve as an arena for comparing different STO algorithms. The source code of the benchmark suite is available at //github.com/XmingHsueh/STOP.
Multilingual machine translation models can benefit from synergy between different language pairs, but also suffer from interference. While there is a growing number of sophisticated methods that aim to eliminate interference, our understanding of interference as a phenomenon is still limited. This work identifies the main factors that contribute to interference in multilingual machine translation. Through systematic experimentation, we find that interference (or synergy) are primarily determined by model size, data size, and the proportion of each language pair within the total dataset. We observe that substantial interference occurs mainly when the model is very small with respect to the available training data, and that using standard transformer configurations with less than one billion parameters largely alleviates interference and promotes synergy. Moreover, we show that tuning the sampling temperature to control the proportion of each language pair in the data is key to balancing the amount of interference between low and high resource language pairs effectively, and can lead to superior performance overall.
We present evidence that language models can learn meaning despite being trained only to perform next token prediction on text, specifically a corpus of programs. Each program is preceded by a specification in the form of (textual) input-output examples. Working with programs enables us to precisely define concepts relevant to meaning in language (e.g., correctness and semantics), making program synthesis well-suited as an intermediate testbed for characterizing the presence (or absence) of meaning in language models. We first train a Transformer model on the corpus of programs, then probe the trained model's hidden states as it completes a program given a specification. Despite providing no inductive bias toward learning the semantics of the language, we find that a linear probe is able to extract abstractions of both current and future program states from the model states. Moreover, there is a strong, statistically significant correlation between the accuracy of the probe and the model's ability to generate a program that implements the specification. To evaluate whether the semantics are represented in the model states rather than learned by the probe, we design a novel experimental procedure that intervenes on the semantics of the language while preserving the lexicon and syntax. We also demonstrate that the model learns to generate correct programs that are, on average, shorter than those in the training set, which is evidence that language model outputs may differ from the training distribution in semantically meaningful ways. In summary, this paper does not propose any new techniques for training language models, but develops an experimental framework for and provides insights into the acquisition and representation of (formal) meaning in language models.
Hyperbolic manifolds for visual representation learning allow for effective learning of semantic class hierarchies by naturally embedding tree-like structures with low distortion within a low-dimensional representation space. The highly separable semantic class hierarchies produced by hyperbolic learning have shown to be powerful in low-shot tasks, however, their application in self-supervised learning is yet to be explored fully. In this work, we explore the use of hyperbolic representation space for self-supervised representation learning for prototype-based clustering approaches. First, we extend the Masked Siamese Networks to operate on the Poincar\'e ball model of hyperbolic space, secondly, we place prototypes on the ideal boundary of the Poincar\'e ball. Unlike previous methods we project to the hyperbolic space at the output of the encoder network and utilise a hyperbolic projection head to ensure that the representations used for downstream tasks remain hyperbolic. Empirically we demonstrate the ability of these methods to perform comparatively to Euclidean methods in lower dimensions for linear evaluation tasks, whilst showing improvements in extreme few-shot learning tasks.
Seeking the equivalent entities among multi-source Knowledge Graphs (KGs) is the pivotal step to KGs integration, also known as \emph{entity alignment} (EA). However, most existing EA methods are inefficient and poor in scalability. A recent summary points out that some of them even require several days to deal with a dataset containing 200,000 nodes (DWY100K). We believe over-complex graph encoder and inefficient negative sampling strategy are the two main reasons. In this paper, we propose a novel KG encoder -- Dual Attention Matching Network (Dual-AMN), which not only models both intra-graph and cross-graph information smartly, but also greatly reduces computational complexity. Furthermore, we propose the Normalized Hard Sample Mining Loss to smoothly select hard negative samples with reduced loss shift. The experimental results on widely used public datasets indicate that our method achieves both high accuracy and high efficiency. On DWY100K, the whole running process of our method could be finished in 1,100 seconds, at least 10* faster than previous work. The performances of our method also outperform previous works across all datasets, where Hits@1 and MRR have been improved from 6% to 13%.
We present self-supervised geometric perception (SGP), the first general framework to learn a feature descriptor for correspondence matching without any ground-truth geometric model labels (e.g., camera poses, rigid transformations). Our first contribution is to formulate geometric perception as an optimization problem that jointly optimizes the feature descriptor and the geometric models given a large corpus of visual measurements (e.g., images, point clouds). Under this optimization formulation, we show that two important streams of research in vision, namely robust model fitting and deep feature learning, correspond to optimizing one block of the unknown variables while fixing the other block. This analysis naturally leads to our second contribution -- the SGP algorithm that performs alternating minimization to solve the joint optimization. SGP iteratively executes two meta-algorithms: a teacher that performs robust model fitting given learned features to generate geometric pseudo-labels, and a student that performs deep feature learning under noisy supervision of the pseudo-labels. As a third contribution, we apply SGP to two perception problems on large-scale real datasets, namely relative camera pose estimation on MegaDepth and point cloud registration on 3DMatch. We demonstrate that SGP achieves state-of-the-art performance that is on-par or superior to the supervised oracles trained using ground-truth labels.
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