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We consider the setting of online convex optimization (OCO) with \textit{exp-concave} losses. The best regret bound known for this setting is $O(n\log{}T)$, where $n$ is the dimension and $T$ is the number of prediction rounds (treating all other quantities as constants and assuming $T$ is sufficiently large), and is attainable via the well-known Online Newton Step algorithm (ONS). However, ONS requires on each iteration to compute a projection (according to some matrix-induced norm) onto the feasible convex set, which is often computationally prohibitive in high-dimensional settings and when the feasible set admits a non-trivial structure. In this work we consider projection-free online algorithms for exp-concave and smooth losses, where by projection-free we refer to algorithms that rely only on the availability of a linear optimization oracle (LOO) for the feasible set, which in many applications of interest admits much more efficient implementations than a projection oracle. We present an LOO-based ONS-style algorithm, which using overall $O(T)$ calls to a LOO, guarantees in worst case regret bounded by $\widetilde{O}(n^{2/3}T^{2/3})$ (ignoring all quantities except for $n,T$). However, our algorithm is most interesting in an important and plausible low-dimensional data scenario: if the gradients (approximately) span a subspace of dimension at most $\rho$, $\rho << n$, the regret bound improves to $\widetilde{O}(\rho^{2/3}T^{2/3})$, and by applying standard deterministic sketching techniques, both the space and average additional per-iteration runtime requirements are only $O(\rho{}n)$ (instead of $O(n^2)$). This improves upon recently proposed LOO-based algorithms for OCO which, while having the same state-of-the-art dependence on the horizon $T$, suffer from regret/oracle complexity that scales with $\sqrt{n}$ or worse.

We consider the idealized setting of gradient flow on the population risk for infinitely wide two-layer ReLU neural networks (without bias), and study the effect of symmetries on the learned parameters and predictors. We first describe a general class of symmetries which, when satisfied by the target function $f^*$ and the input distribution, are preserved by the dynamics. We then study more specific cases. When $f^*$ is odd, we show that the dynamics of the predictor reduces to that of a (non-linearly parameterized) linear predictor, and its exponential convergence can be guaranteed. When $f^*$ has a low-dimensional structure, we prove that the gradient flow PDE reduces to a lower-dimensional PDE. Furthermore, we present informal and numerical arguments that suggest that the input neurons align with the lower-dimensional structure of the problem.

In many complex systems, whether biological or artificial, the thermodynamic costs of communication among their components are large. These systems also tend to split information transmitted between any two components across multiple channels. A common hypothesis is that such inverse multiplexing strategies reduce total thermodynamic costs. So far, however, there have been no physics-based results supporting this hypothesis. This gap existed partially because we have lacked a theoretical framework that addresses the interplay of thermodynamics and information in off-equilibrium systems at any spatiotemporal scale. Here we present the first study that rigorously combines such a framework, stochastic thermodynamics, with Shannon information theory. We develop a minimal model that captures the fundamental features common to a wide variety of communication systems. We find that the thermodynamic cost in this model is a convex function of the channel capacity, the canonical measure of the communication capability of a channel. We also find that this function is not always monotonic, in contrast to previous results not derived from first principles physics. These results clarify when and how to split a single communication stream across multiple channels. In particular, we present Pareto fronts that reveal the trade-off between thermodynamic costs and channel capacity when inverse multiplexing. Due to the generality of our model, our findings could help explain empirical observations of how thermodynamic costs of information transmission make inverse multiplexing energetically favorable in many real-world communication systems.

Neural network training is usually accomplished by solving a non-convex optimization problem using stochastic gradient descent. Although one optimizes over the networks parameters, the main loss function generally only depends on the realization of the neural network, i.e. the function it computes. Studying the optimization problem over the space of realizations opens up new ways to understand neural network training. In particular, usual loss functions like mean squared error and categorical cross entropy are convex on spaces of neural network realizations, which themselves are non-convex. Approximation capabilities of neural networks can be used to deal with the latter non-convexity, which allows us to establish that for sufficiently large networks local minima of a regularized optimization problem on the realization space are almost optimal. Note, however, that each realization has many different, possibly degenerate, parametrizations. In particular, a local minimum in the parametrization space needs not correspond to a local minimum in the realization space. To establish such a connection, inverse stability of the realization map is required, meaning that proximity of realizations must imply proximity of corresponding parametrizations. We present pathologies which prevent inverse stability in general, and, for shallow networks, proceed to establish a restricted space of parametrizations on which we have inverse stability w.r.t. to a Sobolev norm. Furthermore, we show that by optimizing over such restricted sets, it is still possible to learn any function which can be learned by optimization over unrestricted sets.

In this paper, we study an intelligent reflecting surface (IRS) assisted communication system with single-antenna transmitter and receiver, under imperfect channel state information (CSI). More specifically, we deal with the robust selection of binary (on/off) states of the IRS elements in order to maximize the worst-case energy efficiency (EE), given a bounded CSI uncertainty, while satisfying a minimum signal-to-noise ratio (SNR). The IRS phase shifts are adjusted so as to maximize the ideal SNR (i.e., without CSI error), based only on the estimated channels. First, we derive a closed-form expression of the worst-case SNR, and then formulate the robust (discrete) optimization problem. Moreover, we design and analyze a dynamic programming (DP) algorithm that is theoretically guaranteed to achieve the global maximum with polynomial complexity $O(L \log L)$, where $L$ is the number of IRS elements. Finally, numerical simulations confirm the theoretical results. In particular, the proposed algorithm shows identical performance with the exhaustive search, and significantly outperforms a baseline scheme, namely, the activation of all IRS elements.

Salt and pepper noise removal is a common inverse problem in image processing. Traditional denoising methods have two limitations. First, noise characteristics are often not described accurately. For example, the noise location information is often ignored and the sparsity of the salt and pepper noise is often described by L1 norm, which cannot illustrate the sparse variables clearly. Second, conventional methods separate the contaminated image into a recovered image and a noise part, thus resulting in recovering an image with unsatisfied smooth parts and detail parts. In this study, we introduce a noise detection strategy to determine the position of the noise, and a non-convex sparsity regularization depicted by Lp quasi-norm is employed to describe the sparsity of the noise, thereby addressing the first limitation. The morphological component analysis framework with stationary Framelet transform is adopted to decompose the processed image into cartoon, texture, and noise parts to resolve the second limitation. Then, the alternating direction method of multipliers (ADMM) is employed to solve the proposed model. Finally, experiments are conducted to verify the proposed method and compare it with some current state-of-the-art denoising methods. The experimental results show that the proposed method can remove salt and pepper noise while preserving the details of the processed image.

Gradient methods have become mainstream techniques for Bi-Level Optimization (BLO) in learning fields. The validity of existing works heavily rely on either a restrictive Lower- Level Strong Convexity (LLSC) condition or on solving a series of approximation subproblems with high accuracy or both. In this work, by averaging the upper and lower level objectives, we propose a single loop Bi-level Averaged Method of Multipliers (sl-BAMM) for BLO that is simple yet efficient for large-scale BLO and gets rid of the limited LLSC restriction. We further provide non-asymptotic convergence analysis of sl-BAMM towards KKT stationary points, and the comparative advantage of our analysis lies in the absence of strong gradient boundedness assumption, which is always required by others. Thus our theory safely captures a wider variety of applications in deep learning, especially where the upper-level objective is quadratic w.r.t. the lower-level variable. Experimental results demonstrate the superiority of our method.

Monte Carlo Tree Search (MCTS) is a sampling best-first method to search for optimal decisions. The success of MCTS depends heavily on how the MCTS statistical tree is built and the selection policy plays a fundamental role in this. A particular selection policy that works particularly well, widely adopted in MCTS, is the Upper Confidence Bounds for Trees, referred to as UCT. Other more sophisticated bounds have been proposed by the community with the goal to improve MCTS performance on particular problems. Thus, it is evident that while the MCTS UCT behaves generally well, some variants might behave better. As a result of this, multiple works have been proposed to evolve a selection policy to be used in MCTS. Although all these works are inspiring, none of them have carried out an in-depth analysis shedding light under what circumstances an evolved alternative of MCTS UCT might be beneficial in MCTS due to focusing on a single type of problem. In sharp contrast to this, in this work we use five functions of different nature, going from a unimodal function, covering multimodal functions to deceptive functions. We demonstrate how the evolution of the MCTS UCT might be beneficial in multimodal and deceptive scenarios, whereas the MCTS UCT is robust in unimodal scenarios and competitive in the rest of the scenarios used in this study.

Communication and computation are often viewed as separate tasks. This approach is very effective from the perspective of engineering as isolated optimizations can be performed. On the other hand, there are many cases where the main interest is a function of the local information at the devices instead of the local information itself. For such scenarios, information theoretical results show that harnessing the interference in a multiple-access channel for computation, i.e., over-the-air computation (OAC), can provide a significantly higher achievable computation rate than the one with the separation of communication and computation tasks. Besides, the gap between OAC and separation in terms of computation rate increases with more participating nodes. Given this motivation, in this study, we provide a comprehensive survey on practical OAC methods. After outlining fundamentals related to OAC, we discuss the available OAC schemes with their pros and cons. We provide an overview of the enabling mechanisms for achieving reliable computation in the wireless channel. Finally, we summarize the potential applications of OAC and point out some future directions.

The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.