In this paper, it is shown, for the first time, that centralized performance is achievable in decentralized learning without sharing the local datasets. Specifically, when clients adopt an empirical risk minimization with relative-entropy regularization (ERM-RER) learning framework and a forward-backward communication between clients is established, it suffices to share the locally obtained Gibbs measures to achieve the same performance as that of a centralized ERM-RER with access to all the datasets. The core idea is that the Gibbs measure produced by client~$k$ is used, as reference measure, by client~$k+1$. This effectively establishes a principled way to encode prior information through a reference measure. In particular, achieving centralized performance in the decentralized setting requires a specific scaling of the regularization factors with the local sample sizes. Overall, this result opens the door to novel decentralized learning paradigms that shift the collaboration strategy from sharing data to sharing the local inductive bias via the reference measures over the set of models.

It is well known that this is unlearnable, but we sketch a proof here.

Logistic regression is commonly used for modeling dichotomous outcomes.

Modifications to a neural network's input and output layers are often required to accommodate the specificities of most practical learning tasks.

We study square loss in a realizable time-series framework with martingale difference noise. Our main result is a fast rate excess risk bound which shows that whenever a trajectory hypercontractivity condition holds, the risk of the leastsquares estimator on dependent data matches the iid rate order-wise after a burn-in time. In comparison, many existing results in learning from dependent data have rates where the effective sample size is deflated by a factor of the mixing-time of the underlying process, even after the burn-in time. Furthermore, our results allow the covariate process to exhibit long range correlations which are substantially weaker than geometric ergodicity.

We study square loss in a realizable time-series framework with martingale difference noise. Our main result is a fast rate excess risk bound which shows that whenever a trajectory hypercontractivity condition holds, the risk of the least-squares estimator on dependent data matches the iid rate order-wise after a burn-in time. In comparison, many existing results in learning from dependent data have rates where the effective sample size is deflated by a factor of the mixing-time of the underlying process, even after the burn-in time. Furthermore, our results allow the covariate process to exhibit long range correlations which are substantially weaker than geometric ergodicity. We call this phenomenon learning with little mixing, and present several examples for when it occurs: bounded function classes for which the $L^2$ and $L^{2+\epsilon}$ norms are equivalent, finite state irreducible and aperiodic Markov chains, various parametric models, and a broad family of infinite dimensional $\ell^2(\mathbb{N})$ ellipsoids. By instantiating our main result to system identification of nonlinear dynamics with generalized linear model transitions, we obtain a nearly minimax optimal excess risk bound after only a polynomial burn-in time.

The ability to design molecules while preserving similarity to a target molecule and/or property is crucial for various applications in drug discovery, chemical design, and biology. We introduce in this paper an efficient training-free method for navigating and sampling from the molecular space with a generative Chemical Language Model (CLM), while using the molecular similarity to the target as a guide. Our method leverages the contextual representations learned from the CLM itself to estimate the molecular similarity, which is then used to adjust the autoregressive sampling strategy of the CLM. At each step of the decoding process, the method tracks the distance of the current generations from the target and updates the logits to encourage the preservation of similarity in generations. We implement the method using a recently proposed $\sim$47M parameter SMILES-based CLM, GP-MoLFormer, and therefore refer to the method as GP-MoLFormer-Sim, which enables a test-time update of the deep generative policy to reflect the contextual similarity to a set of guide molecules. The method is further integrated into a genetic algorithm (GA) and tested on a set of standard molecular optimization benchmarks involving property optimization, molecular rediscovery, and structure-based drug design. Results show that, GP-MoLFormer-Sim, combined with GA (GP-MoLFormer-Sim+GA) outperforms existing training-free baseline methods, when the oracle remains black-box. The findings in this work are a step forward in understanding and guiding the generative mechanisms of CLMs.