We develop an algorithmic framework for solving convex optimization problems using no-regret game dynamics. By converting the problem of minimizing a convex function into an auxiliary problem of solving a min-max game in a sequential fashion, we can consider a range of strategies for each of the two-players who must select their actions one after the other. A common choice for these strategies are so-called no-regret learning algorithms, and we describe a number of such and prove bounds on their regret. We then show that many classical first-order methods for convex optimization -- including average-iterate gradient descent, the Frank-Wolfe algorithm, Nesterov's acceleration methods, and the accelerated proximal method -- can be interpreted as special cases of our framework as long as each player makes the correct choice of no-regret strategy. Proving convergence rates in this framework becomes very straightforward, as they follow from plugging in the appropriate known regret bounds. Our framework also gives rise to a number of new first-order methods for special cases of convex optimization that were not previously known.
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Algorithmic stablecoins (AS) are one special type of stablecoins that are not backed by any asset (equiv. without collateral). They stand to revolutionize the way a sovereign fiat operates. As implemented, these coins are poorly stabilized in most cases, easily deviating from the price target or even falling into a catastrophic collapse (a.k.a. Death spiral), and are as a result dismissed as a Ponzi scheme. However, is this the whole picture? In this paper, we try to reveal the truth and clarify such a deceptive concept. We find that Ponzi is basically a financial protocol that pays existing investors with funds collected from new ones. Running a Ponzi, however, does not necessarily imply that any participant is in any sense losing out, as long as the game can be perpetually rolled over. Economists call such realization as a \textit{rational Ponzi game}. We thereby propose a rational model in the context of AS and draw its holding conditions. We apply the model to examine: \textit{whether or not the algorithmic stablecoin is a rational Ponzi game.} Accordingly, we discuss two types of algorithmic stablecoins (\text{Rebase} \& \text{Seigniorage shares}) and dig into the historical market performance of two impactful projects (\text{Ampleforth} \& \text{TerraUSD}, respectively) to demonstrate the effectiveness of our model.
Gradient Boosting Machines (GBMs) have demonstrated remarkable success in solving diverse problems by utilizing Taylor expansions in functional space. However, achieving a balance between performance and generality has posed a challenge for GBMs. In particular, gradient descent-based GBMs employ the first-order Taylor expansion to ensure applicability to all loss functions, while Newton's method-based GBMs use positive Hessian information to achieve superior performance at the expense of generality. To address this issue, this study proposes a new generic Gradient Boosting Machine called Trust-region Boosting (TRBoost). In each iteration, TRBoost uses a constrained quadratic model to approximate the objective and applies the Trust-region algorithm to solve it and obtain a new learner. Unlike Newton's method-based GBMs, TRBoost does not require the Hessian to be positive definite, thereby allowing it to be applied to arbitrary loss functions while still maintaining competitive performance similar to second-order algorithms. The convergence analysis and numerical experiments conducted in this study confirm that TRBoost is as general as first-order GBMs and yields competitive results compared to second-order GBMs. Overall, TRBoost is a promising approach that balances performance and generality, making it a valuable addition to the toolkit of machine learning practitioners.
Given the ubiquity of non-separable optimization problems in real worlds, in this paper we analyze and extend the large-scale version of the well-known cooperative coevolution (CC), a divide-and-conquer optimization framework, on non-separable functions. First, we reveal empirical reasons of why decomposition-based methods are preferred or not in practice on some non-separable large-scale problems, which have not been clearly pointed out in many previous CC papers. Then, we formalize CC to a continuous game model via simplification, but without losing its essential property. Different from previous evolutionary game theory for CC, our new model provides a much simpler but useful viewpoint to analyze its convergence, since only the pure Nash equilibrium concept is needed and more general fitness landscapes can be explicitly considered. Based on convergence analyses, we propose a hierarchical decomposition strategy for better generalization, as for any decomposition there is a risk of getting trapped into a suboptimal Nash equilibrium. Finally, we use powerful distributed computing to accelerate it under the multi-level learning framework, which combines the fine-tuning ability from decomposition with the invariance property of CMA-ES. Experiments on a set of high-dimensional functions validate both its search performance and scalability (w.r.t. CPU cores) on a clustering computing platform with 400 CPU cores.
This paper explores equilibrium concepts for Bayesian games, which are fundamental models of games with incomplete information. We aim at three desirable properties of equilibria. First, equilibria can be naturally realized by introducing a mediator into games. Second, an equilibrium can be computed efficiently in a distributed fashion. Third, any equilibrium in that class approximately maximizes social welfare, as measured by the price of anarchy, for a broad class of games. These three properties allow players to compute an equilibrium and realize it via a mediator, thereby settling into a stable state with approximately optimal social welfare. Our main result is the existence of an equilibrium concept that satisfies these three properties. Toward this goal, we characterize various (non-equivalent) extensions of correlated equilibria, collectively known as Bayes correlated equilibria. In particular, we focus on communication equilibria (also known as coordination mechanisms), which can be realized by a mediator who gathers each player's private information and then sends correlated recommendations to the players. We show that if each player minimizes a variant of regret called untruthful swap regret in repeated play of Bayesian games, the empirical distribution of these dynamics converges to a communication equilibrium. We present an efficient algorithm for minimizing untruthful swap regret with a sublinear upper bound, which we prove to be tight up to a multiplicative constant. As a result, by simulating the dynamics with our algorithm, we can efficiently compute an approximate communication equilibrium. Furthermore, we extend existing lower bounds on the price of anarchy based on the smoothness arguments from Bayes Nash equilibria to equilibria obtained by the proposed dynamics.
Nonsmooth composite optimization with orthogonality constraints has a broad spectrum of applications in statistical learning and data science. However, this problem is generally challenging to solve due to its non-convex and non-smooth nature. Existing solutions are limited by one or more of the following restrictions: (i) they are full gradient methods that require high computational costs in each iteration; (ii) they are not capable of solving general nonsmooth composite problems; (iii) they are infeasible methods and can only achieve the feasibility of the solution at the limit point; (iv) they lack rigorous convergence guarantees; (v) they only obtain weak optimality of critical points. In this paper, we propose \textit{\textbf{OBCD}}, a new Block Coordinate Descent method for solving general nonsmooth composite problems under Orthogonality constraints. \textit{\textbf{OBCD}} is a feasible method with low computation complexity footprints. In each iteration, our algorithm updates $k$ rows of the solution matrix ($k\geq2$ is a parameter) to preserve the constraints. Then, it solves a small-sized nonsmooth composite optimization problem under orthogonality constraints either exactly or approximately. We demonstrate that any exact block-$k$ stationary point is always an approximate block-$k$ stationary point, which is equivalent to the critical stationary point. We are particularly interested in the case where $k=2$ as the resulting subproblem reduces to a one-dimensional nonconvex problem. We propose a breakpoint searching method and a fifth-order iterative method to solve this problem efficiently and effectively. We also propose two novel greedy strategies to find a good working set to further accelerate the convergence of \textit{\textbf{OBCD}}. Finally, we have conducted extensive experiments on several tasks to demonstrate the superiority of our approach.
Inferring unknown constraints is a challenging and crucial problem in many robotics applications. When only expert demonstrations are available, it becomes essential to infer the unknown domain constraints to deploy additional agents effectively. In this work, we propose an approach to infer affine constraints in control tasks after observing expert demonstrations. We formulate the constraint inference problem as an inverse optimization problem, and we propose an alternating optimization scheme that infers the unknown constraints by minimizing a KKT residual objective. We demonstrate the effectiveness of our method in a number of simulations, and show that our method can infer less conservative constraints than a recent baseline method while maintaining comparable safety guarantees.
Emotion recognition in conversation (ERC) aims to detect the emotion label for each utterance. Motivated by recent studies which have proven that feeding training examples in a meaningful order rather than considering them randomly can boost the performance of models, we propose an ERC-oriented hybrid curriculum learning framework. Our framework consists of two curricula: (1) conversation-level curriculum (CC); and (2) utterance-level curriculum (UC). In CC, we construct a difficulty measurer based on "emotion shift" frequency within a conversation, then the conversations are scheduled in an "easy to hard" schema according to the difficulty score returned by the difficulty measurer. For UC, it is implemented from an emotion-similarity perspective, which progressively strengthens the model's ability in identifying the confusing emotions. With the proposed model-agnostic hybrid curriculum learning strategy, we observe significant performance boosts over a wide range of existing ERC models and we are able to achieve new state-of-the-art results on four public ERC datasets.
This manuscript portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.
This paper surveys and organizes research works in a new paradigm in natural language processing, which we dub "prompt-based learning". Unlike traditional supervised learning, which trains a model to take in an input x and predict an output y as P(y|x), prompt-based learning is based on language models that model the probability of text directly. To use these models to perform prediction tasks, the original input x is modified using a template into a textual string prompt x' that has some unfilled slots, and then the language model is used to probabilistically fill the unfilled information to obtain a final string x, from which the final output y can be derived. This framework is powerful and attractive for a number of reasons: it allows the language model to be pre-trained on massive amounts of raw text, and by defining a new prompting function the model is able to perform few-shot or even zero-shot learning, adapting to new scenarios with few or no labeled data. In this paper we introduce the basics of this promising paradigm, describe a unified set of mathematical notations that can cover a wide variety of existing work, and organize existing work along several dimensions, e.g.the choice of pre-trained models, prompts, and tuning strategies. To make the field more accessible to interested beginners, we not only make a systematic review of existing works and a highly structured typology of prompt-based concepts, but also release other resources, e.g., a website //pretrain.nlpedia.ai/ including constantly-updated survey, and paperlist.
The growing energy and performance costs of deep learning have driven the community to reduce the size of neural networks by selectively pruning components. Similarly to their biological counterparts, sparse networks generalize just as well, if not better than, the original dense networks. Sparsity can reduce the memory footprint of regular networks to fit mobile devices, as well as shorten training time for ever growing networks. In this paper, we survey prior work on sparsity in deep learning and provide an extensive tutorial of sparsification for both inference and training. We describe approaches to remove and add elements of neural networks, different training strategies to achieve model sparsity, and mechanisms to exploit sparsity in practice. Our work distills ideas from more than 300 research papers and provides guidance to practitioners who wish to utilize sparsity today, as well as to researchers whose goal is to push the frontier forward. We include the necessary background on mathematical methods in sparsification, describe phenomena such as early structure adaptation, the intricate relations between sparsity and the training process, and show techniques for achieving acceleration on real hardware. We also define a metric of pruned parameter efficiency that could serve as a baseline for comparison of different sparse networks. We close by speculating on how sparsity can improve future workloads and outline major open problems in the field.
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