Riccardo Grazzi

Vladimir R. Kostic, Pietro Novelli, Riccardo Grazzi et al.

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 learning the transfer operator or the generator of the system, which 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 learning problem. 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.

Riccardo Grazzi, Arya Akhavan, John Isak Texas Falk et al.

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.

Julien Siems, Riccardo Grazzi, Kirill Kalinin et al.

Over the last years, state-tracking tasks, particularly permutation composition, have become a testbed to understand the limits of sequence models architectures like Transformers and RNNs (linear and non-linear). However, these are often sequence-to-sequence tasks: learning to map actions (permutations) to states, which is incompatible with the next-token prediction setting commonly used to train language models. We address this gap by converting permutation composition into code via REPL traces that interleave state-reveals through prints and variable transformations. We show that linear RNNs capable of state-tracking excel also in this setting, while Transformers still fail. Motivated by this representation, we investigate why tracking states in code is generally difficult: actions are not always fully observable. We frame this as tracking the state of a probabilistic finite-state automaton with deterministic state reveals and show that linear RNNs can be worse than non-linear RNNs at tracking states in this setup.

Advait Gadhikar, Riccardo Grazzi, James Hensman

The presence of outliers in Large Language Models (LLMs) weights and activations makes them difficult to quantize. Recent work has leveraged rotations to mitigate these outliers. In this work, we propose methods that learn fusible rotations by minimizing principled and cheap proxy objectives to the weight quantization error. We primarily focus on GPTQ as the quantization method. Our main method is OptRot, which reduces weight outliers simply by minimizing the element-wise fourth power of the rotated weights. We show that OptRot outperforms both Hadamard rotations and more expensive, data-dependent methods like SpinQuant and OSTQuant for weight quantization. It also improves activation quantization in the W4A8 setting. We also propose a data-dependent method, OptRot$^{+}$, that further improves performance by incorporating information on the activation covariance. In the W4A4 setting, we see that both OptRot and OptRot$^{+}$ perform worse, highlighting a trade-off between weight and activation quantization.

Riccardo Grazzi, Julien Siems, Simon Schrodi et al.

State of the art foundation models such as GPT-4 perform surprisingly well at in-context learning (ICL), a variant of meta-learning concerning the learned ability to solve tasks during a neural network forward pass, exploiting contextual information provided as input to the model. This useful ability emerges as a side product of the foundation model's massive pretraining. While transformer models are currently the state of the art in ICL, this work provides empirical evidence that Mamba, a newly proposed state space model which scales better than transformers w.r.t. the input sequence length, has similar ICL capabilities. 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 closely 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 long input sequences. This is an exciting finding in meta-learning and may enable generalizations of in-context learned AutoML algorithms (like TabPFN or Optformer) to long input sequences.

Riccardo Grazzi, Julien Siems, Arber Zela et al.

Linear Recurrent Neural Networks (LRNNs) such as Mamba, RWKV, GLA, mLSTM, and DeltaNet have emerged as efficient alternatives to Transformers for long sequences. However, both Transformers and LRNNs struggle to perform state-tracking, which may impair performance in tasks such as code evaluation. In one forward pass, current architectures are unable to solve even parity, the simplest state-tracking task, which non-linear RNNs can handle effectively. 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 such as DeltaNet. We prove that finite precision LRNNs with state-transition matrices having only positive eigenvalues cannot solve parity, while non-triangular matrices 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 experiments confirm that extending the eigenvalue range of Mamba and DeltaNet to include negative values not only enables them to solve parity but consistently improves their performance on state-tracking tasks. We also show that state-tracking enabled LRNNs can be pretrained stably and efficiently at scale (1.3B parameters), achieving competitive performance on language modeling and showing promise on code and math tasks.

Julien Siems, Timur Carstensen, Arber Zela et al.

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. Diagonal matrices, used in models such as Mamba, GLA, or mLSTM, yield fast runtime but have limited expressivity. To address this, recent architectures such as DeltaNet and RWKV-7 adopted a diagonal plus rank--1 structure, which allows simultaneous token and channel mixing, improving associative recall and, as recently shown, state-tracking when allowing state-transition matrices to have negative eigenvalues. 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. We provide a detailed theoretical characterization of the state-tracking capability of DeltaProduct in finite precision, showing how it improves by increasing $n_h$. Our extensive experiments demonstrate that DeltaProduct outperforms DeltaNet in both state-tracking and language modeling, while also showing significantly improved length extrapolation capabilities.

Riccardo Grazzi, Massimiliano Pontil, Saverio Salzo

We study the problem of efficiently computing the derivative of the fixed-point of a parametric nondifferentiable contraction map. This problem has wide applications in machine learning, including hyperparameter optimization, meta-learning and data poisoning attacks. We analyze two popular approaches: iterative differentiation (ITD) and approximate implicit differentiation (AID). A key challenge behind the nonsmooth setting is that the chain rule does not hold anymore. We build upon the work by Bolte et al. (2022), who prove linear convergence of nonsmooth ITD under a piecewise Lipschitz smooth assumption. In the deterministic case, we provide a linear rate for AID and an improved linear rate for ITD which closely match the ones for the smooth setting. We further introduce NSID, a new stochastic method to compute the implicit derivative when the contraction map is defined as the composition of an outer map and an inner map which is accessible only through a stochastic unbiased estimator. We establish rates for the convergence of NSID, encompassing the best available rates in the smooth setting. We also present illustrative experiments confirming our analysis.

Riccardo Grazzi, Massimiliano Pontil, Saverio Salzo

We analyse 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, equilibrium models, 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. However, there are situations, e.g., meta learning and equilibrium models, in which the warm-start procedure is not well-suited or ineffective. In this work we show that without warm-start, it is still possible to achieve order-wise (near) optimal sample complexity. 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 $ε$-stationary point using $O(ε^{-2})$ and $\tilde{O}(ε^{-1})$ samples for the stochastic and the deterministic setting, respectively. Finally, compared to methods using warm-start, our approach yields a simpler analysis that does not need to study the coupled interactions between the upper-level and lower-level iterates.

Riccardo Grazzi, Valentin Flunkert, David Salinas et al.

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 ofthe 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.

Riccardo Grazzi, Massimiliano Pontil, Saverio Salzo

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 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. The method that we propose is a stochastic variant of the approximate implicit differentiation approach in (Pedregosa, 2016). We provide 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. In particular, our main bound is agnostic to the choice of the two stochastic solvers employed by the procedure. We provide numerical experiments to support our theoretical analysis and to show the advantage of using stochastic hypergradients in practice.

Riccardo Grazzi, Luca Franceschi, Massimiliano Pontil et al.

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.

Giulia Denevi, Carlo Ciliberto, Riccardo Grazzi et al.

We study the problem of learning-to-learn: inferring a learning algorithm that works well on tasks sampled from an unknown distribution. As class of algorithms we consider Stochastic Gradient Descent on the true risk regularized by the square euclidean distance to a bias vector. We present an average excess risk bound for such a learning algorithm. This result quantifies the potential benefit of using a bias vector with respect to the unbiased case. We then address the problem of estimating the bias from a sequence of tasks. We propose a meta-algorithm which incrementally updates the bias, as new tasks are observed. The low space and time complexity of this approach makes it appealing in practice. We provide guarantees on the learning ability of the meta-algorithm. A key feature of our results is that, when the number of tasks grows and their variance is relatively small, our learning-to-learn approach has a significant advantage over learning each task in isolation by Stochastic Gradient Descent without a bias term. We report on numerical experiments which demonstrate the effectiveness of our approach.

Luca Franceschi, Riccardo Grazzi, Massimiliano Pontil et al.

In (Franceschi et al., 2018) we proposed a unified mathematical framework, grounded on bilevel programming, that encompasses gradient-based hyperparameter optimization and meta-learning. We formulated an approximate version of the problem where the inner objective is solved iteratively, and gave sufficient conditions ensuring convergence to the exact problem. In this work we show how to optimize learning rates, automatically weight the loss of single examples and learn hyper-representations with Far-HO, a software package based on the popular deep learning framework TensorFlow that allows to seamlessly tackle both HO and ML problems.

Luca Franceschi, Paolo Frasconi, Saverio Salzo et al.

We introduce a framework based on bilevel programming that unifies gradient-based hyperparameter optimization and meta-learning. We show that an approximate version of the bilevel problem can be solved by taking into explicit account the optimization dynamics for the inner objective. Depending on the specific setting, the outer variables take either the meaning of hyperparameters in a supervised learning problem or parameters of a meta-learner. We provide sufficient conditions under which solutions of the approximate problem converge to those of the exact problem. We instantiate our approach for meta-learning in the case of deep learning where representation layers are treated as hyperparameters shared across a set of training episodes. In experiments, we confirm our theoretical findings, present encouraging results for few-shot learning and contrast the bilevel approach against classical approaches for learning-to-learn.