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The Gromov--Hausdorff distance measures the difference in shape between compact metric spaces and poses a notoriously difficult problem in combinatorial optimization. We introduce its quadratic relaxation over a convex polytope whose solutions provably deliver the Gromov--Hausdorff distance. The optimality guarantee is enabled by the fact that the search space of our approach is not constrained to a generalization of bijections, unlike in other relaxations such as the Gromov--Wasserstein distance. We suggest the Frank--Wolfe algorithm with $O(n^3)$-time iterations for solving the relaxation and numerically demonstrate its performance on metric spaces of hundreds of points. In particular, we obtain a new upper bound of the Gromov--Hausdorff distance between the unit circle and the unit hemisphere equipped with Euclidean metric. Our approach is implemented as a Python package dGH.

Next-generation wireless communication systems impose much stricter requirements for transmission rate, latency, and reliability. The peak data rate of 6G networks should be no less than 1 Tb/s, which is comparable to existing long-haul optical transport networks. It is believed that using long error-correcting codes (ECC) with soft-decision decoding (SDD) is not feasible in this case due to the resulting high power consumption. On the other hand, ECC with hard-decision decoding (HDD) suffers from significant performance degradation. In this paper, we consider a concatenated solution consisting of an outer long HDD code and an inner short SDD code. The latter code is a crucial component of the system and the focus of our research. Due to its short length, the code cannot correct all errors, but it is designed to minimize the number of errors. Such codes are known as error-reducing codes. We investigate the error-reducing properties of superposition codes. Initially, we explore sparse regression codes (SPARCs) with Gaussian signals. This approach outperforms error-reducing binary LDPC codes optimized by Barakatain, et al. (2018) in terms of performance but faces limitations in practical applicability due to high implementation complexity. Subsequently, we propose an LDPC-based superposition code scheme with low-complexity soft successive interference cancellation (SIC) decoding. This scheme demonstrates comparable performance to SPARCs while maintaining manageable complexity. Numerical results were obtained for inner codes with an overhead (OH) of 8.24% within a concatenated scheme (15% OH) with an outer hard-decision decoded staircase code (6.25% OH).

Robust distributed learning with Byzantine failures has attracted extensive research interests in recent years. However, most of existing methods suffer from curse of dimensionality, which is increasingly serious with the growing complexity of modern machine learning models. In this paper, we design a new method that is suitable for high dimensional problems, under arbitrary number of Byzantine attackers. The core of our design is a direct high dimensional semi-verified mean estimation method. Our idea is to identify a subspace first. The components of mean value perpendicular to this subspace can be estimated via gradient vectors uploaded from worker machines, while the components within this subspace are estimated using auxiliary dataset. We then use our new method as the aggregator of distributed learning problems. Our theoretical analysis shows that the new method has minimax optimal statistical rates. In particular, the dependence on dimensionality is significantly improved compared with previous works.

This work studies post-training parameter quantization in large language models (LLMs). We introduce quantization with incoherence processing (QuIP), a new method based on the insight that quantization benefits from incoherent weight and Hessian matrices, i.e., from the weights and the directions in which it is important to round them accurately being unaligned with the coordinate axes. QuIP consists of two steps: (1) an adaptive rounding procedure minimizing a quadratic proxy objective; (2) efficient pre- and post-processing that ensures weight and Hessian incoherence via multiplication by random orthogonal matrices. We complement QuIP with the first theoretical analysis for an LLM-scale quantization algorithm, and show that our theory also applies to an existing method, OPTQ. Empirically, we find that our incoherence preprocessing improves several existing quantization algorithms and yields the first LLM quantization methods that produce viable results using only two bits per weight. Our code can be found at //github.com/jerry-chee/QuIP .

We study the query complexity of geodesically convex (g-convex) optimization on a manifold. To isolate the effect of that manifold's curvature, we primarily focus on hyperbolic spaces. In a variety of settings (smooth or not; strongly g-convex or not; high- or low-dimensional), known upper bounds worsen with curvature. It is natural to ask whether this is warranted, or an artifact. For many such settings, we propose a first set of lower bounds which indeed confirm that (negative) curvature is detrimental to complexity. To do so, we build on recent lower bounds (Hamilton and Moitra, 2021; Criscitiello and Boumal, 2022) for the particular case of smooth, strongly g-convex optimization. Using a number of techniques, we also secure lower bounds which capture dependence on condition number and optimality gap, which was not previously the case. We suspect these bounds are not optimal. We conjecture optimal ones, and support them with a matching lower bound for a class of algorithms which includes subgradient descent, and a lower bound for a related game. Lastly, to pinpoint the difficulty of proving lower bounds, we study how negative curvature influences (and sometimes obstructs) interpolation with g-convex functions.

A posteriori error estimates based on residuals can be used for reliable error control of numerical methods. Here, we consider them in the context of ordinary differential equations and Runge-Kutta methods. In particular, we take the approach of Dedner & Giesselmann (2016) and investigate it when used to select the time step size. We focus on step size control stability when combined with explicit Runge-Kutta methods and demonstrate that a standard I controller is unstable while more advanced PI and PID controllers can be designed to be stable. We compare the stability properties of residual-based estimators and classical error estimators based on an embedded Runge-Kutta method both analytically and in numerical experiments.

The field of fine-grained complexity aims at proving conditional lower bounds on the time complexity of computational problems. One of the most popular assumptions, Strong Exponential Time Hypothesis (SETH), implies that SAT cannot be solved in $2^{(1-\epsilon)n}$ time. In recent years, it has been proved that known algorithms for many problems are optimal under SETH. Despite the wide applicability of SETH, for many problems, there are no known SETH-based lower bounds, so the quest for new reductions continues. Two barriers for proving SETH-based lower bounds are known. Carmosino et al. (ITCS 2016) introduced the Nondeterministic Strong Exponential Time Hypothesis (NSETH) stating that TAUT cannot be solved in time $2^{(1-\epsilon)n}$ even if one allows nondeterminism. They used this hypothesis to show that some natural fine-grained reductions would be difficult to obtain: proving that, say, 3-SUM requires time $n^{1.5+\epsilon}$ under SETH, breaks NSETH and this, in turn, implies strong circuit lower bounds. Recently, Belova et al. (SODA 2023) introduced the so-called polynomial formulations to show that for many NP-hard problems, proving any explicit exponential lower bound under SETH also implies strong circuit lower bounds. We prove that for a range of problems from P, including $k$-SUM and triangle detection, proving superlinear lower bounds under SETH is challenging as it implies new circuit lower bounds. To this end, we show that these problems can be solved in nearly linear time with oracle calls to evaluating a polynomial of constant degree. Then, we introduce a strengthening of SETH stating that solving SAT in time $2^{(1-\varepsilon)n}$ is difficult even if one has constant degree polynomial evaluation oracle calls. This hypothesis is stronger and less believable than SETH, but refuting it is still challenging: we show that this implies circuit lower bounds.

Adam is a commonly used stochastic optimization algorithm in machine learning. However, its convergence is still not fully understood, especially in the non-convex setting. This paper focuses on exploring hyperparameter settings for the convergence of vanilla Adam and tackling the challenges of non-ergodic convergence related to practical application. The primary contributions are summarized as follows: firstly, we introduce precise definitions of ergodic and non-ergodic convergence, which cover nearly all forms of convergence for stochastic optimization algorithms. Meanwhile, we emphasize the superiority of non-ergodic convergence over ergodic convergence. Secondly, we establish a weaker sufficient condition for the ergodic convergence guarantee of Adam, allowing a more relaxed choice of hyperparameters. On this basis, we achieve the almost sure ergodic convergence rate of Adam, which is arbitrarily close to $o(1/\sqrt{K})$. More importantly, we prove, for the first time, that the last iterate of Adam converges to a stationary point for non-convex objectives. Finally, we obtain the non-ergodic convergence rate of $O(1/K)$ for function values under the Polyak-Lojasiewicz (PL) condition. These findings build a solid theoretical foundation for Adam to solve non-convex stochastic optimization problems.

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.

Federated learning is a new distributed machine learning framework, where a bunch of heterogeneous clients collaboratively train a model without sharing training data. In this work, we consider a practical and ubiquitous issue in federated learning: intermittent client availability, where the set of eligible clients may change during the training process. Such an intermittent client availability model would significantly deteriorate the performance of the classical Federated Averaging algorithm (FedAvg for short). We propose a simple distributed non-convex optimization algorithm, called Federated Latest Averaging (FedLaAvg for short), which leverages the latest gradients of all clients, even when the clients are not available, to jointly update the global model in each iteration. Our theoretical analysis shows that FedLaAvg attains the convergence rate of $O(1/(N^{1/4} T^{1/2}))$, achieving a sublinear speedup with respect to the total number of clients. We implement and evaluate FedLaAvg with the CIFAR-10 dataset. The evaluation results demonstrate that FedLaAvg indeed reaches a sublinear speedup and achieves 4.23% higher test accuracy than FedAvg.