Deploying complex machine learning models on resource-constrained devices is challenging due to limited computational power, memory, and model retrainability. To address these limitations, a hybrid system can be established by augmenting the local model with a server-side model, where samples are selectively deferred by a rejector and then sent to the server for processing. The hybrid system enables efficient use of computational resources while minimizing the overhead associated with server usage. The recently proposed Learning to Help (L2H) model trains a server model given a fixed local (client) model, differing from the Learning to Defer (L2D) framework, which trains the client for a fixed (expert) server. In both L2D and L2H, the training includes learning a rejector at the client to determine when to query the server. In this work, we extend the L2H model from binary to multi-class classification problems and demonstrate its applicability in a number of different scenarios of practical interest in which access to the server may be limited by cost, availability, or policy. We derive a stage-switching surrogate loss function that is differentiable, convex, and consistent with the Bayes rule corresponding to the 0-1 loss for the L2H model. Experiments show that our proposed methods offer an efficient and practical solution for multi-class classification in resource-constrained environments.

Training a deep neural network often involves stochastic optimization, meaning each run will produce a different model. The seed used to initialize random elements of the optimization procedure heavily influences the quality of a trained model, which may be obscure from many commonly reported summary statistics, like accuracy. However, random seed is often not included in hyper-parameter optimization, perhaps because the relationship between seed and model quality is hard to describe. This work attempts to describe the relationship between deep net models trained with different random seeds and the behavior of the expected model. We adopt robust hypothesis testing to propose a novel summary statistic for network similarity, referred to as the $\alpha$-trimming level. We use the $\alpha$-trimming level to show that the empirical cumulative distribution function of an ensemble model created from a collection of trained models with different random seeds approximates the average of these functions as the number of models in the collection grows large. This insight provides guidance for how many random seeds should be sampled to ensure that an ensemble of these trained models is a reliable representative. We also show that the $\alpha$-trimming level is more expressive than different performance metrics like validation accuracy, churn, or expected calibration error when taken alone and may help with random seed selection in a more principled fashion. We demonstrate the value of the proposed statistic in real experiments and illustrate the advantage of fine-tuning over random seed with an experiment in transfer learning.

ABSTRACT Training a deep neural network (DNN) often involves stochastic optimization, which means each run will produce a different model. Several works suggest this variability is negligible when models have the same performance, which in the case of classification is test accuracy. However, models with similar test accuracy may not be computing the same function. We propose a new measure of closeness between classification models based on the output of the network before thresholding. Our measure is based on a robust hypothesis-testing framework and can be adapted to other quantities derived from trained models.

Recent works have shown that imposing tensor structures on the coefficient tensor in regression problems can lead to more reliable parameter estimation and lower sample complexity compared to vector-based methods. This work investigates a new low-rank tensor model, called Low Separation Rank (LSR), in Generalized Linear Model (GLM) problems. The LSR model -- which generalizes the well-known Tucker and CANDECOMP/PARAFAC (CP) models, and is a special case of the Block Tensor Decomposition (BTD) model -- is imposed onto the coefficient tensor in the GLM model. This work proposes a block coordinate descent algorithm for parameter estimation in LSR-structured tensor GLMs. Most importantly, it derives a minimax lower bound on the error threshold on estimating the coefficient tensor in LSR tensor GLM problems. The minimax bound is proportional to the intrinsic degrees of freedom in the LSR tensor GLM problem, suggesting that its sample complexity may be significantly lower than that of vectorized GLMs. This result can also be specialised to lower bound the estimation error in CP and Tucker-structured GLMs. The derived bounds are comparable to tight bounds in the literature for Tucker linear regression, and the tightness of the minimax lower bound is further assessed numerically. Finally, numerical experiments on synthetic datasets demonstrate the efficacy of the proposed LSR tensor model for three regression types (linear, logistic and Poisson). Experiments on a collection of medical imaging datasets demonstrate the usefulness of the LSR model over other tensor models (Tucker and CP) on real, imbalanced data with limited available samples.

Training deep neural networks (DNNs) used in modern machine learning is computationally expensive. Machine learning scientists, therefore, rely on stochastic first-order methods for training, coupled with significant hand-tuning, to obtain good performance. To better understand performance variability of different stochastic algorithms, including second-order methods, we conduct an empirical study that treats performance as a response variable across multiple training sessions of the same model. Using 2-factor Analysis of Variance (ANOVA) with interactions, we show that batch size used during training has a statistically significant effect on the peak accuracy of the methods, and that full batch largely performed the worst. In addition, we found that second-order optimizers (SOOs) generally exhibited significantly lower variance at specific batch sizes, suggesting they may require less hyperparameter tuning, leading to a reduced overall time to solution for model training.

This paper considers the problem of matrix-variate logistic regression. The fundamental error threshold on estimating coefficient matrices in the logistic regression problem is found by deriving a lower bound on the minimax risk. The focus of this paper is on derivation of a minimax risk lower bound for low-rank coefficient matrices. The bound depends explicitly on the dimensions and distribution of the covariates, the rank and energy of the coefficient matrix, and the number of samples. The resulting bound is proportional to the intrinsic degrees of freedom in the problem, which suggests the sample complexity of the low-rank matrix logistic regression problem can be lower than that for vectorized logistic regression. \color{red}\color{black} The proof techniques utilized in this work also set the stage for development of minimax lower bounds for tensor-variate logistic regression problems.

We study the best-arm identification problem in multi-armed bandits with stochastic, potentially private rewards, when the goal is to identify the arm with the highest quantile at a fixed, prescribed level. First, we propose a (non-private) successive elimination algorithm for strictly optimal best-arm identification, we show that our algorithm is $\delta$-PAC and we characterize its sample complexity. Further, we provide a lower bound on the expected number of pulls, showing that the proposed algorithm is essentially optimal up to logarithmic factors. Both upper and lower complexity bounds depend on a special definition of the associated suboptimality gap, designed in particular for the quantile bandit problem, as we show when the gap approaches zero, best-arm identification is impossible. Second, motivated by applications where the rewards are private, we provide a differentially private successive elimination algorithm whose sample complexity is finite even for distributions with infinite support-size, and we characterize its sample complexity as well. Our algorithms do not require prior knowledge of either the suboptimality gap or other statistical information related to the bandit problem at hand.

Blind source separation algorithms such as independent component analysis (ICA) are widely used in the analysis of neuroimaging data. In order to leverage larger sample sizes, different data holders/sites may wish to collaboratively learn feature representations. However, such datasets are often privacy-sensitive, precluding centralized analyses that pool the data at a single site. A recently proposed algorithm uses message-passing between sites and a central aggregator to perform a decentralized joint ICA (djICA) without sharing the data. However, this method does not satisfy formal privacy guarantees. We propose a differentially private algorithm for performing ICA in a decentralized data setting. Differential privacy provides a formal and mathematically rigorous privacy guarantee by introducing noise into the messages. Conventional approaches to decentralized differentially private algorithms may require too much noise due to the typically small sample sizes at each site. We leverage a recently proposed correlated noise protocol to remedy the excessive noise problem of the conventional schemes. We investigate the performance of the proposed algorithm on synthetic and real fMRI datasets to show that our algorithm outperforms existing approaches and can sometimes reach the same level of utility as the corresponding non-private algorithm. This indicates that it is possible to have meaningful utility while preserving privacy.

We provide high probability sample complexity guarantees for non-parametric structure learning of tree-shaped graphical models whose nodes are discrete random variables with a finite or countable alphabet, both in the noiseless and noisy regimes. First, we introduce a new, fundamental quantity called the (noisy) information threshold, which arises naturally from the error analysis of the Chow-Liu algorithm and characterizes not only the sample complexity, but also the inherent impact of the noise on the structure learning task, without explicit assumptions on the distribution of the model. This allows us to present the first non-parametric, high-probability finite sample complexity bounds on tree-structure learning from potentially noise-corrupted data. In particular, for number of nodes $p$, success rate $1-\delta$, and a fixed value of the information threshold, our sample complexity bounds for exact structure recovery are of the order of $\mathcal{O}\big(\log^{1+\zeta} (p/\delta)\big) $, for all $\zeta>0$, for both noiseless and noisy settings. Subsequently, we apply our results on two classes of hidden models, namely, the $M$-ary erasure channel and the generalized symmetric channel, illustrating the usefulness and importance of our framework. As a byproduct of our analysis, this paper resolves the open problem of tree structure learning in the presence of non-identically distributed observation noise, providing explicit conditions on the convergence of the Chow-Liu algorithm under this setting as well.

Many applications of machine learning, such as human health research, involve processing private or sensitive information. Privacy concerns may impose significant hurdles to collaboration in scenarios where there are multiple sites holding data and the goal is to estimate properties jointly across all datasets. Differentially private decentralized algorithms can provide strong privacy guarantees. However, the accuracy of the joint estimates may be poor when the datasets at each site are small. This paper proposes a new framework, Correlation Assisted Private Estimation (CAPE), for designing privacy-preserving decentralized algorithms with better accuracy guarantees in an honest-but-curious model. CAPE can be used in conjunction with the functional mechanism for statistical and machine learning optimization problems. A tighter characterization of the functional mechanism is provided that allows CAPE to achieve the same performance as a centralized algorithm in the decentralized setting using all datasets. Empirical results on regression and neural network problems for both synthetic and real datasets show that differentially private methods can be competitive with non-private algorithms in many scenarios of interest.