Linear Recurrent Neural Networks (linear RNNs) have emerged as competitive alternatives to Transformers for sequence modeling, offering efficient training and linear-time inference. However, existing architectures face a fundamental trade-off between expressivity and efficiency, dictated by the structure of their state-transition matrices. While diagonal matrices used in architectures like Mamba, GLA, or mLSTM yield fast runtime, they suffer from severely limited expressivity. To address this, recent architectures such as (Gated) DeltaNet and RWKVv7 adopted a diagonal plus rank-1 structure, allowing simultaneous token-channel mixing, which overcomes some expressivity limitations with only a slight decrease in training efficiency. Building on the interpretation of DeltaNet's recurrence as performing one step of online gradient descent per token on an associative recall loss, we introduce DeltaProduct, which instead takes multiple ($n_h$) steps per token. This naturally leads to diagonal plus rank-$n_h$ state-transition matrices, formed as products of $n_h$ generalized Householder transformations, providing a tunable mechanism to balance expressivity and efficiency and a stable recurrence. Through extensive experiments, we demonstrate that DeltaProduct achieves superior state-tracking and language modeling capabilities while exhibiting significantly improved length extrapolation compared to DeltaNet. Additionally, we also strengthen the theoretical foundation of DeltaNet's expressivity by proving that it can solve dihedral group word problems in just two layers.

Linear Recurrent Neural Networks (LRNNs) such as Mamba, RWKV, GLA, mLSTM, and DeltaNet have emerged as efficient alternatives to Transformers in large language modeling, offering linear scaling with sequence length and improved training efficiency. However, LRNNs struggle to perform state-tracking which may impair performance in tasks such as code evaluation or tracking a chess game. Even parity, the simplest state-tracking task, which non-linear RNNs like LSTM handle effectively, cannot be solved by current LRNNs. Recently, Sarrof et al. (2024) demonstrated that the failure of LRNNs like Mamba to solve parity stems from restricting the value range of their diagonal state-transition matrices to $[0, 1]$ and that incorporating negative values can resolve this issue. We extend this result to non-diagonal LRNNs, which have recently shown promise in models such as DeltaNet. We prove that finite precision LRNNs with state-transition matrices having only positive eigenvalues cannot solve parity, while complex eigenvalues are needed to count modulo $3$. Notably, we also prove that LRNNs can learn any regular language when their state-transition matrices are products of identity minus vector outer product matrices, each with eigenvalues in the range $[-1, 1]$. Our empirical results confirm that extending the eigenvalue range of models like Mamba and DeltaNet to include negative values not only enables them to solve parity but consistently improves their performance on state-tracking tasks. Furthermore, pre-training LRNNs with an extended eigenvalue range for language modeling achieves comparable performance and stability while showing promise on code and math data. Our work enhances the expressivity of modern LRNNs, broadening their applicability without changing the cost of training or inference.

Important examples are given by hyperparameter optimization and meta-learning (Franceschi et al., 2018; Lee et al., 2019), where (1) expresses the optimality conditions of a lower-level minimization problem. Further examples include learning a surrogate model for data poisoning attacks (Xiao et al., 2015; Muñoz-González et al., 2017), deep equilibrium models (Bai et al., 2019) or OptNet (Amos & Kolter, 2017). All these problems may present nonsmooth mappings Φ. For instance, consider hyperparameter optimization or data poisoning attacks for SVMs, or meta-learning for image classification, where Φ is evaluated through the forward pass of a neural net with RELU activations (Bertinetto et al., 2019; Lee et al., 2019; Rajeswaran et al., 2019). In addition, when such settings are applied to large datasets, evaluating the map Φ would be too costly, but we can usually apply stochastic methods through the composite stochastic structure in (2), where only T involves a computation on the full training set (e.g., a gradient descent step).

This work provides empirical evidence that Mamba, a newly proposed selective structured state space model, has similar in-context learning (ICL) capabilities as transformers. We evaluated Mamba on tasks involving simple function approximation as well as more complex natural language processing problems. Our results demonstrate that across both categories of tasks, Mamba matches the performance of transformer models for ICL. Further analysis reveals that like transformers, Mamba appears to solve ICL problems by incrementally optimizing its internal representations. Overall, our work suggests that Mamba can be an efficient alternative to transformers for ICL tasks involving longer input sequences. Recent advancements in large-scale neural language modeling (Brown et al., 2020) have demonstrated that Transformer models (Vaswani et al., 2017) exhibit in-context learning (ICL) capabilities: after (self-supervised) pre-training, they can infer how to perform tasks only from input examples without the need for explicit training nor fine-tuning.

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learn the transfer operator of the system, that in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the context of transfer operator regression. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

We study the linear contextual bandit problem where an agent has to select one candidate from a pool and each candidate belongs to a sensitive group. In this setting, candidates' rewards may not be directly comparable between groups, for example when the agent is an employer hiring candidates from different ethnic groups and some groups have a lower reward due to discriminatory bias and/or social injustice. We propose a notion of fairness that states that the agent's policy is fair when it selects a candidate with highest relative rank, which measures how good the reward is when compared to candidates from the same group. This is a very strong notion of fairness, since the relative rank is not directly observed by the agent and depends on the underlying reward model and on the distribution of rewards. Thus we study the problem of learning a policy which approximates a fair policy under the condition that the contexts are independent between groups and the distribution of rewards of each group is absolutely continuous. In particular, we design a greedy policy which at each round constructs a ridge regression estimate from the observed context-reward pairs, and then computes an estimate of the relative rank of each candidate using the empirical cumulative distribution function. We prove that, despite its simplicity and the lack of an initial exploration phase, the greedy policy achieves, up to log factors and with high probability, a fair pseudo-regret of order $\sqrt{dT}$ after $T$ rounds, where $d$ is the dimension of the context vectors. The policy also satisfies demographic parity at each round when averaged over all possible information available before the selection. Finally, we use simulated settings and experiments on the US census data to show that our policy achieves sub-linear fair pseudo-regret also in practice.

We analyze a general class of bilevel problems, in which the upper-level problem consists in the minimization of a smooth objective function and the lower-level problem is to find the fixed point of a smooth contraction map. This type of problems include instances of meta-learning, hyperparameter optimization and data poisoning adversarial attacks. Several recent works have proposed algorithms which warm-start the lower-level problem, i.e. they use the previous lower-level approximate solution as a staring point for the lower-level solver. This warm-start procedure allows one to improve the sample complexity in both the stochastic and deterministic settings, achieving in some cases the order-wise optimal sample complexity. We show that without warm-start, it is still possible to achieve order-wise optimal and near-optimal sample complexity for the stochastic and deterministic settings, respectively. In particular, we propose a simple method which uses stochastic fixed point iterations at the lower-level and projected inexact gradient descent at the upper-level, that reaches an $\epsilon$-stationary point using $O(\epsilon^{-2})$ and $\tilde{O}(\epsilon^{-1})$ samples for the stochastic and the deterministic setting, respectively. Compared to methods using warm-start, ours is better suited for meta-learning and yields a simpler analysis that does not need to study the coupled interactions between the upper-level and lower-level iterates.

While classical time series forecasting considers individual time series in isolation, recent advances based on deep learning showed that jointly learning from a large pool of related time series can boost the forecasting accuracy. However, the accuracy of these methods suffers greatly when modeling out-of-sample time series, significantly limiting their applicability compared to classical forecasting methods. To bridge this gap, we adopt a meta-learning view of the time series forecasting problem. We introduce a novel forecasting method, called Meta Global-Local Auto-Regression (Meta-GLAR), that adapts to each time series by learning in closed-form the mapping from the representations produced by a recurrent neural network (RNN) to one-step-ahead forecasts. Crucially, the parameters of the RNN are learned across multiple time series by backpropagating through the closed-form adaptation mechanism. In our extensive empirical evaluation we show that our method is competitive with the state-of-the-art in out-of-sample forecasting accuracy reported in earlier work.

Bilevel optimization problems are receiving increasing attention in machine learning as they provide a natural framework for hyperparameter optimization and meta-learning. A key step to tackle these problems in the design of optimization algorithms for bilevel optimization is the efficient computation of the gradient of the upper-level objective (hypergradient). In this work, we study stochastic approximation schemes for the hypergradient, which are important when the lower-level problem is empirical risk minimization on a large dataset. We provide iteration complexity bounds for the mean square error of the hypergradient approximation, under the assumption that the lower-level problem is accessible only through a stochastic mapping which is a contraction in expectation. Preliminary numerical experiments support our theoretical analysis.

We study a general class of bilevel problems, consisting in the minimization of an upper-level objective which depends on the solution to a parametric fixed-point equation. Important instances arising in machine learning include hyperparameter optimization, meta-learning, and certain graph and recurrent neural networks. Typically the gradient of the upper-level objective (hypergradient) is hard or even impossible to compute exactly, which has raised the interest in approximation methods. We investigate some popular approaches to compute the hypergradient, based on reverse mode iterative differentiation and approximate implicit differentiation. Under the hypothesis that the fixed point equation is defined by a contraction mapping, we present a unified analysis which allows for the first time to quantitatively compare these methods, providing explicit bounds for their iteration complexity. This analysis suggests a hierarchy in terms of computational efficiency among the above methods, with approximate implicit differentiation based on conjugate gradient performing best. We present an extensive experimental comparison among the methods which confirm the theoretical findings.