Entropy maximization and free energy minimization are general physics principles for modeling dynamic systems. Notable examples include modeling decision-making within the brain using the free-energy principle, optimizing the accuracy-complexity trade-off when accessing hidden variables with the information bottleneck principle (Tishby et al. 2000), and navigation in random environments using information maximization (Vergassola et al. 2007). Building on this principle, we propose a new class of bandit algorithms that maximize an approximation to the information of a key variable within the system. To this end, we develop an approximated, analytical physics-based representation of the entropy to forecast the information gain of each action and greedily choose the one with the largest information gain. This method yields strong performances in classical bandit settings. Motivated by its empirical success, we prove its asymptotic optimality for the multi-armed bandit problem with Gaussian rewards. Since it encompasses the system’s properties in a single, global functional, this approach can be efficiently adapted to more complex bandit settings. This calls for further investigation of information maximization approaches for multi-armed bandit problems.

Understanding generalization of overparametrized models remains a fundamental challenge in machine learning. The literature mostly studies generalization from an interpolation point of view, taking convergence towards a global minimum of the training loss for granted. This interpolation paradigm does not seem valid for complex tasks such as in-context learning or diffusion. It has instead been empirically observed that the trained models go from global minima to spurious local minima of the training loss as the number of training samples becomes larger than some level we call optimization threshold. This paper explores theoretically this phenomenon in the context of two-layer ReLU networks. We demonstrate that, despite overparametrization, networks might converge towards simpler solutions rather than interpolating training data, which leads to a drastic improvement on the test loss. Our analysis relies on the so called early alignment phase, during which neurons align toward specific directions. This directional alignment leads to a simplicity bias, wherein the network approximates the ground truth model without converging to the global minimum of the training loss. Our results suggest this bias, resulting in an optimization threshold from which interpolation is not reached anymore, is beneficial and enhances the generalization of trained models.

The framework of uncoupled online learning in multiplayer games has made significant progress in recent years. In particular, the development of time-varying games has considerably expanded its modeling capabilities. However, current regret bounds quickly become vacuous when the game undergoes significant variations over time, even when these variations are easy to predict. Intuitively, the ability of players to forecast future payoffs should lead to tighter guarantees, yet existing approaches fail to incorporate this aspect. This work aims to fill this gap by introducing a novel prediction-aware framework for time-varying games, where agents can forecast future payoffs and adapt their strategies accordingly. In this framework, payoffs depend on an underlying state of nature that agents predict in an online manner. To leverage these predictions, we propose the POMWU algorithm, a contextual extension of the optimistic Multiplicative Weight Update algorithm, for which we establish theoretical guarantees on social welfare and convergence to equilibrium. Our results demonstrate that, under bounded prediction errors, the proposed framework achieves performance comparable to the static setting. Finally, we empirically demonstrate the effectiveness of POMWU in a traffic routing experiment.

We study the dynamics of gradient flow with small weight decay on general training losses $F: \mathbb{R}^d \to \mathbb{R}$. Under mild regularity assumptions and assuming convergence of the unregularised gradient flow, we show that the trajectory with weight decay $\lambda$ exhibits a two-phase behaviour as $\lambda \to 0$. During the initial fast phase, the trajectory follows the unregularised gradient flow and converges to a manifold of critical points of $F$. Then, at time of order $1/\lambda$, the trajectory enters a slow drift phase and follows a Riemannian gradient flow minimising the $\ell_2$-norm of the parameters. This purely optimisation-based phenomenon offers a natural explanation for the \textit{grokking} effect observed in deep learning, where the training loss rapidly reaches zero while the test loss plateaus for an extended period before suddenly improving. We argue that this generalisation jump can be attributed to the slow norm reduction induced by weight decay, as explained by our analysis. We validate this mechanism empirically on several synthetic regression tasks.

Training neural networks with first order optimisation methods is at the core of the empirical success of deep learning. The scale of initialisation is a crucial factor, as small initialisations are generally associated to a feature learning regime, for which gradient descent is implicitly biased towards simple solutions. This work provides a general and quantitative description of the early alignment phase, originally introduced by Maennel et al. (2018). For small initialisation and one hidden ReLU layer networks, the early stage of the training dynamics leads to an alignment of the neurons towards key directions. This alignment induces a sparse representation of the network, which is directly related to the implicit bias of gradient flow at convergence. This sparsity inducing alignment however comes at the expense of difficulties in minimising the training objective: we also provide a simple data example for which overparameterised networks fail to converge towards global minima and only converge to a spurious stationary point instead.

Multiplayer bandits have recently garnered significant attention due to their relevance in cognitive radio networks. While the existing body of literature predominantly focuses on synchronous players, real-world radio networks, such as those in IoT applications, often feature asynchronous (i.e., randomly activated) devices. This highlights the need for addressing the more challenging asynchronous multiplayer bandits problem. Our first result shows that a natural extension of UCB achieves a minimax regret of $\mathcal{O}(\sqrt{T\log(T)})$ in the centralized setting. More significantly, we introduce Cautious Greedy, which uses $\mathcal{O}(\log(T))$ communications and whose instance-dependent regret is constant if the optimal policy assigns at least one player to each arm (a situation proven to occur when arm means are sufficiently close). Otherwise, the regret is, as usual, $\log(T)$ times the sum of some inverse sub-optimality gaps. We substantiate the optimality of Cautious Greedy through lower-bound analysis based on data-dependent terms. Therefore, we establish a strong baseline for asynchronous multiplayer bandits, at least with $\mathcal{O}(\log(T))$ communications.

Due to its empirical success in few-shot classification and reinforcement learning, meta-learning has recently received significant interest. Meta-learning methods leverage data from previous tasks to learn a new task in a sample-efficient manner. In particular, model-agnostic methods look for initialization points from which gradient descent quickly adapts to any new task. Although it has been empirically suggested that such methods perform well by learning shared representations during pretraining, there is limited theoretical evidence of such behavior. More importantly, it has not been shown that these methods still learn a shared structure, despite architectural misspecifications. In this direction, this work shows, in the limit of an infinite number of tasks, that first-order ANIL with a linear two-layer network architecture successfully learns linear shared representations. This result even holds with _overparametrization_; having a width larger than the dimension of the shared representations results in an asymptotically low-rank solution. The learned solution then yields a good adaptation performance on any new task after a single gradient step. Overall, this illustrates how well model-agnostic methods such as first-order ANIL can learn shared representations.

This work considers a repeated principal-agent bandit game, where the principal can only interact with her environment through the agent. The principal and the agent have misaligned objectives and the choice of action is only left to the agent. However, the principal can influence the agent's decisions by offering incentives which add up to his rewards. The principal aims to iteratively learn an incentive policy to maximize her own total utility. This framework extends usual bandit problems and is motivated by several practical applications, such as healthcare or ecological taxation, where traditionally used mechanism design theories often overlook the learning aspect of the problem. We present nearly optimal (with respect to a horizon $T$) learning algorithms for the principal's regret in both multi-armed and linear contextual settings. Finally, we support our theoretical guarantees through numerical experiments.

Collaborative learning offers a promising avenue for leveraging decentralized data. However, collaboration in groups of strategic learners is not a given. In this work, we consider strategic agents who wish to train a model together but have sampling distributions of different quality. The collaboration is organized by a benevolent aggregator who gathers samples so as to maximize total welfare, but is unaware of data quality. This setting allows us to shed light on the deleterious effect of adverse selection in collaborative learning. More precisely, we demonstrate that when data quality indices are private, the coalition may undergo a phenomenon known as unravelling, wherein it shrinks up to the point that it becomes empty or solely comprised of the worst agent. We show how this issue can be addressed without making use of external transfers, by proposing a novel method inspired by probabilistic verification. This approach makes the grand coalition a Nash equilibrium with high probability despite information asymmetry, thereby breaking unravelling.

In Economics, the concept of externality refers to any indirect effect resulting from an interaction between players and affecting a third party without compensation. Most of the models within which externality has been studied assume that agents have perfect knowledge of their environment and preferences. This is a major hindrance to the practical implementation of many proposed solutions. To adress this issue, we consider a two-players bandit game setting where the actions of one of the player affect the other one. Building upon this setup, we extend the Coase theorem [Coase, 2013], which suggests that the optimal approach for maximizing the social welfare in the presence of externality is to establish property rights, i.e., enabling transfers and bargaining between the players. Nonetheless, this fundamental result relies on the assumption that bargainers possess perfect knowledge of the underlying game. We first demonstrate that in the absence of property rights in the considered online scenario, the social welfare breaks down. We then provide a policy for the players, which allows them to learn a bargaining strategy which maximizes the total welfare, recovering the Coase theorem under uncertainty.

Due mostly to its application to cognitive radio networks, multiplayer bandits gained a lot of interest in the last decade. A considerable progress has been made on its theoretical aspect. However, the current algorithms are far from applicable and many obstacles remain between these theoretical results and a possible implementation of multiplayer bandits algorithms in real communication networks. This survey contextualizes and organizes the rich multiplayer bandits literature. In light of the existing works, some clear directions for future research appear. We believe that a further study of these different directions might lead to theoretical algorithms adapted to real-world situations.

Controlling the parameters' norm often yields good generalisation when training neural networks. Beyond simple intuitions, the relation between regularising parameters' norm and obtained estimators remains theoretically misunderstood. For one hidden ReLU layer networks with unidimensional data, this work shows the parameters' norm required to represent a function is given by the total variation of its second derivative, weighted by a $\sqrt{1+x^2}$ factor. Notably, this weighting factor disappears when the norm of bias terms is not regularised. The presence of this additional weighting factor is of utmost significance as it is shown to enforce the uniqueness and sparsity (in the number of kinks) of the minimal norm interpolator. Conversely, omitting the bias' norm allows for non-sparse solutions. Penalising the bias terms in the regularisation, either explicitly or implicitly, thus leads to sparse estimators.

Potential buyers of a product or service, before making their decisions, tend to read reviews written by previous consumers. We consider Bayesian consumers with heterogeneous preferences, who sequentially decide whether to buy an item of unknown quality, based on previous buyers’ reviews. The quality is multi-dimensional and may occasionally vary over time; the reviews are also multi-dimensional. In the simple uni-dimensional and static setting, beliefs about the quality are known to converge to its true value. Our paper extends this result in several ways. First, a multi-dimensional quality is considered, second, rates of convergence are provided, third, a dynamical Markovian model with varying quality is studied. In this dynamical setting the cost of learning is shown to be small.

Multi-task learning leverages structural similarities between multiple tasks to learn despite very few samples. Motivated by the recent success of neural networks applied to data-scarce tasks, we consider a linear low-dimensional shared representation model. Despite an extensive literature, existing theoretical results either guarantee weak estimation rates or require a large number of samples per task. This work provides the first estimation error bound for the trace norm regularized estimator when the number of samples per task is small. The advantages of trace norm regularization for learning data-scarce tasks extend to meta-learning and are confirmed empirically on synthetic datasets.

The training of neural networks by gradient descent methods is a cornerstone of the deep learning revolution. Yet, despite some recent progress, a complete theory explaining its success is still missing. This article presents, for orthogonal input vectors, a precise description of the gradient flow dynamics of training one-hidden layer ReLU neural networks for the mean squared error at small initialisation. In this setting, despite non-convexity, we show that the gradient flow converges to zero loss and characterise its implicit bias towards minimum variation norm. Furthermore, some interesting phenomena are highlighted: a quantitative description of the initial alignment phenomenon and a proof that the process follows a specific saddle to saddle dynamics.

We study online learning for optimal allocation when the resource to be allocated is time. An agent receives task proposals sequentially according to a Poisson process and can either accept or reject a proposed task. If she accepts the proposal, she is busy for the duration of the task and obtains a reward that depends on the task duration. If she rejects it, she remains on hold until a new task proposal arrives. We study the regret incurred by the agent first when she knows her reward function but does not know the distribution of the task duration, and then when she does not know her reward function, either. Faster rates are finally obtained by adding structural assumptions on the distribution of rides or on the reward function. This natural setting bears similarities with contextual (one-armed) bandits, but with the crucial difference that the normalized reward associated to a context depends on the whole distribution of contexts.

Motivated by packet routing in computer networks, online queuing systems are composed of queues receiving packets at different rates. Repeatedly, they send packets to servers, each of them treating only at most one packet at a time. In the centralized case, the number of accumulated packets remains bounded (i.e., the system is stable) as long as the ratio between service rates and arrival rates is larger than $1$. In the decentralized case, individual no-regret strategies ensures stability when this ratio is larger than $2$. Yet, myopically minimizing regret disregards the long term effects due to the carryover of packets to further rounds. On the other hand, minimizing long term costs leads to stable Nash equilibria as soon as the ratio exceeds $\frac{e}{e-1}$. Stability with decentralized learning strategies with a ratio below $2$ was a major remaining question. We first argue that for ratios up to $2$, cooperation is required for stability of learning strategies, as selfish minimization of policy regret, a patient notion of regret, might indeed still be unstable in this case. We therefore consider cooperative queues and propose the first learning decentralized algorithm guaranteeing stability of the system as long as the ratio of rates is larger than $1$, thus reaching performances comparable to centralized strategies.

Strategic information is valuable either by remaining private (for instance if it is sensitive) or, on the other hand, by being used publicly to increase some utility. These two objectives are antagonistic and leaking this information might be more rewarding than concealing it. Unlike classical solutions that focus on the first point, we consider instead agents that optimize a natural trade-off between both objectives.We formalize this as an optimization problem where the objective mapping is regularized by the amount of information revealed to the adversary (measured as a divergence between the prior and posterior on the private knowledge). Quite surprisingly, when combined with the entropic regularization, the Sinkhorn loss naturally emerges in the optimization objective, making it efficiently solvable. We apply these techniques to preserve some privacy in online repeated auctions.

We study a multiplayer stochastic multi-armed bandit problem in which players cannot communicate, and if two or more players pull the same arm, a collision occurs and the involved players receive zero reward. We consider the challenging heterogeneous setting, in which different arms may have different means for different players, and propose a new and efficient algorithm that combines the idea of leveraging forced collisions for implicit communication and that of performing matching eliminations. We present a finite-time analysis of our algorithm, giving the first sublinear minimax regret bound for this problem, and prove that if the optimal assignment of players to arms is unique, our algorithm attains the optimal O(ln(T)) regret, solving an open question raised at NeurIPS 2018 by Bistritz and Leshem (2018).

Motivated by cognitive radios, stochastic multi-player multi-armed bandits gained a lot of interest recently. In this class of problems, several players simultaneously pull arms and encounter a collision – with 0 reward – if some of them pull the same arm at the same time. While the cooperative case where players maximize the collective reward (obediently following some fixed protocol) has been mostly considered, robustness to malicious players is a crucial and challenging concern. Existing approaches consider only the case of adversarial jammers whose objective is to blindly minimize the collective reward. We shall consider instead the more natural class of selfish players whose incentives are to maximize their individual rewards, potentially at the expense of the social welfare. We provide the first algorithm robust to selfish players (a.k.a. Nash equilibrium) with a logarithmic regret, when the arm performance is observed. When collisions are also observed, Grim Trigger type of strategies enable some implicit communication-based algorithms and we construct robust algorithms in two different settings: the homogeneous (with a regret comparable to the centralized optimal one) and heterogeneous cases (for an adapted and relevant notion of regret). We also provide impossibility results when only the reward is observed or when arm means vary arbitrarily among players.

We investigate stochastic combinatorial multi-armed bandit with semi-bandit feedback (CMAB). In CMAB, the question of the existence of an efficient policy with an optimal asymptotic regret (up to a factor poly-logarithmic with the action size) is still open for many families of distributions, including mutually independent outcomes, and more generally the multivariate \emph{sub-Gaussian} family. We propose to answer the above question for these two families by analyzing variants of the Combinatorial Thompson Sampling policy (CTS). For mutually independent outcomes in $[0,1]$, we propose a tight analysis of CTS using Beta priors. We then look at the more general setting of multivariate sub-Gaussian outcomes and propose a tight analysis of CTS using Gaussian priors. This last result gives us an alternative to the Efficient Sampling for Combinatorial Bandit policy (ESCB), which, although optimal, is not computationally efficient.

Plato provides fast approximate analytics on time series, by precomputing and storing compressed time series. Plato's key novelty is the delivery of tight deterministic error guarantees for the linear algebra operators over vectors/time series, the inner product operator and arithmetic operators. Composing them allows for evaluating common statistics, such as correlation and cross-correlation. In the offline processing phase, Plato (i) segments each time series into several disjoint segmentations using known fixed-length or variable-length segmentation algorithms; (ii) compresses each segment by a compression function that is coming from a user-chosen compression function family; and (iii) associates to each segment 1 to 3 precomputed error measures. In the online query processing phase, Plato uses the error measures to compute the error guarantees. Importantly, we identify certain compression function families that lead to theoretically and experimentally higher quality guarantees.

Motivated by cognitive radio networks, we consider the stochastic multiplayer multi-armed bandit problem, where several players pull arms simultaneously and collisions occur if one of them is pulled by several players at the same stage. We present a decentralized algorithm that achieves the same performance as a centralized one, contradicting the existing lower bounds for that problem. This is possible by ``hacking'' the standard model by constructing a communication protocol between players that deliberately enforces collisions, allowing them to share their information at a negligible cost. This motivates the introduction of a more appropriate dynamic setting without sensing, where similar communication protocols are no longer possible. However, we show that the logarithmic growth of the regret is still achievable for this model with a new algorithm.