Symbolic Regression

In this paper we compare and contrast the behavior of the posterior predictive distribution to the risk of the the maximum a posteriori estimator for the random features regression model in the overparameterized regime. We will focus on the variance of the posterior predictive distribution (Bayesian model average) and compare its asymptotics to that of the risk of the MAP estimator. In the regime where the model dimensions grow faster than any constant multiple of the number of samples, asymptotic agreement between these two quantities is governed by the phase transition in the signal-to-noise ratio. They also asymptotically agree with each other when the number of samples grow faster than any constant multiple of model dimensions. Numerical simulations illustrate finer distributional properties of the two quantities for finite dimensions. We conjecture they have Gaussian fluctuations and exhibit similar properties as found by previous authors in a Gaussian sequence model, this is of independent theoretical interest.

Deep reinforcement learning (DRL) has led to a wide range of advances in sequential decision-making tasks. However, the complexity of neural network policies makes it difficult to understand and deploy with limited computational resources. Currently, employing compact symbolic expressions as symbolic policies is a promising strategy to obtain simple and interpretable policies. Previous symbolic policy methods usually involve complex training processes and pre-trained neural network policies, which are inefficient and limit the application of symbolic policies. In this paper, we propose an efficient gradient-based learning method named Efficient Symbolic Policy Learning (ESPL) that learns the symbolic policy from scratch in an end-to-end way. We introduce a symbolic network as the search space and employ a path selector to find the compact symbolic policy. By doing so we represent the policy with a differentiable symbolic expression and train it in an off-policy manner which further improves the efficiency. In addition, in contrast with previous symbolic policies which only work in single-task RL because of complexity, we expand ESPL on meta-RL to generate symbolic policies for unseen tasks. Experimentally, we show that our approach generates symbolic policies with higher performance and greatly improves data efficiency for single-task RL. In meta-RL, we demonstrate that compared with neural network policies the proposed symbolic policy achieves higher performance and efficiency and shows the potential to be interpretable.

We study the problem of collaboratively learning least squares estimates for $m$ agents. Each agent observes a different subset of the features---e.g., containing data collected from sensors of varying resolution. Our goal is to determine how to coordinate the agents in order to produce the best estimator for each agent. We propose a distributed, semi-supervised algorithm Collab, consisting of three steps: local training, aggregation, and distribution. Our procedure does not require communicating the labeled data, making it communication efficient and useful in settings where the labeled data is inaccessible. Despite this handicap, our procedure is nearly asymptotically, local-minimax optimal---even among estimators allowed to communicate the labeled data such as imputation methods. We test our method on US Census data. We also discuss generalizations of our method to non-Gaussian feature settings, non-linear settings, and Federated Learning.

In this paper, we consider the uncertainty quantification problem for regression models. Specifically, we consider an individual calibration objective for characterizing the quantiles of the prediction model. While such an objective is well-motivated from downstream tasks such as newsvendor cost, the existing methods have been largely heuristic and lack of statistical guarantee in terms of individual calibration. We show via simple examples that the existing methods focusing on population-level calibration guarantees such as average calibration or sharpness can lead to harmful and unexpected results. We propose simple nonparametric calibration methods that are agnostic of the underlying prediction model and enjoy both computational efficiency and statistical consistency. Our approach enables a better understanding of the possibility of individual calibration, and we establish matching upper and lower bounds for the calibration error of our proposed methods. Technically, our analysis combines the nonparametric analysis with a covering number argument for parametric analysis, which advances the existing theoretical analyses in the literature of nonparametric density estimation and quantile bandit problems. Importantly, the nonparametric perspective sheds new theoretical insights into regression calibration in terms of the curse of dimensionality and reconciles the existing results on the impossibility of individual calibration. To our knowledge, we make the first effort to reach both individual calibration and finite-sample guarantee with minimal assumptions in terms of conformal prediction. Numerical experiments show the advantage of such a simple approach under various metrics, and also under covariates shift. We hope our work provides a simple benchmark and a starting point of theoretical ground for future research on regression calibration.

Membership inference attacks are designed to determine, using black box access to trained models, whether a particular example was used in training or not. Membership inference can be formalized as a hypothesis testing problem. The most effective existing attacks estimate the distribution of some test statistic (usually the model's confidence on the true label) on points that were (and were not) used in training by training many \emph{shadow models}---i.e. models of the same architecture as the model being attacked, trained on a random subsample of data. While effective, these attacks are extremely computationally expensive, especially when the model under attack is large. \footnotetext[0]{ Martin and Shuai are the lead authors, and other authors are ordered alphabetically. \{maberlop,shuat\}@amazon.com} We introduce a new class of attacks based on performing quantile regression on the distribution of confidence scores induced by the model under attack on points that are not used in training. We show that our method is competitive with state-of-the-art shadow model attacks, while requiring substantially less compute because our attack requires training only a single model. Moreover, unlike shadow model attacks, our proposed attack does not require any knowledge of the architecture of the model under attack and is therefore truly ``black-box". We show the efficacy of this approach in an extensive series of experiments on various datasets and model architectures. Our code is available at \href{https://github.com/amazon-science/quantile-mia}{github.com/amazon-science/quantile-mia.}

We derive upper bounds for random design linear regression with dependent ($\beta$-mixing) data absent any realizability assumptions. In contrast to the strictly realizable martingale noise regime, no sharp \emph{instance-optimal} non-asymptotics are available in the literature. Up to constant factors, our analysis correctly recovers the variance term predicted by the Central Limit Theorem---the noise level of the problem---and thus exhibits graceful degradation as we introduce misspecification. Past a burn-in, our result is sharp in the moderate deviations regime, and in particular does not inflate the leading order term by mixing time factors.

A fundamental challenge in neuro-symbolic AI is to devise primitives that fuse the logical and neural concepts. The Neural Theorem Prover has proposed the notion of soft-unification to turn the symbolic comparison between terms (i.e. unification) into a comparison in embedding space. It has been shown that soft-unification is a powerful mechanism that can be used to learn logic rules in an end-to-end differentiable manner. We study soft-unification from a conceptual point and outline several desirable properties of this operation. These include non-redundancy in the proof, well-defined proof scores, and non-sparse gradients. Unfortunately, these properties are not satisfied by previous systems such as the Neural Theorem Prover. Therefore, we introduce a more principled framework called DeepSoftLog based on probabilistic rather than fuzzy semantics. Our experiments demonstrate that DeepSoftLog can outperform the state-of-the-art on neuro-symbolic benchmarks, highlighting the benefits of these properties.

Decision trees are a well-established tool in machine learning for classification and regression tasks. In this paper, we introduce a novel non-parametric Bayesian model that uses variational inference to approximate a posterior distribution over the space of stochastic decision trees. We evaluate the model's performance on 18 datasets and demonstrate its competitiveness with other state-of-the-art methods in regression tasks. We also explore its application to causal inference problems. We provide a fully vectorized implementation of our algorithm in PyTorch.

With the advent of deep learning, progressively larger neural networks have been designed to solve complex tasks. We take advantage of these capacity-rich models to lower the cost of inference by exploiting computation in superposition. To reduce the computational burden per input, we propose Multiple-Input-Multiple-Output Neural Networks (MIMONets) capable of handling many inputs at once. MIMONets augment various deep neural network architectures with variable binding mechanisms to represent an arbitrary number of inputs in a compositional data structure via fixed-width distributed representations. Accordingly, MIMONets adapt nonlinear neural transformations to process the data structure holistically, leading to a speedup nearly proportional to the number of superposed input items in the data structure. After processing in superposition, an unbinding mechanism recovers each transformed input of interest. MIMONets also provide a dynamic trade-off between accuracy and throughput by an instantaneous on-demand switching between a set of accuracy-throughput operating points, yet within a single set of fixed parameters. We apply the concept of MIMONets to both CNN and Transformer architectures resulting in MIMOConv and MIMOFormer, respectively. Empirical evaluations show that MIMOConv achieves $\approx 2$–$4\times$ speedup at an accuracy delta within [+0.68, -3.18]% compared to WideResNet CNNs on CIFAR10 and CIFAR100. Similarly, MIMOFormer can handle $2$–$4$ inputs at once while maintaining a high average accuracy within a [-1.07, -3.43]% delta on the long range arena benchmark. Finally, we provide mathematical bounds on the interference between superposition channels in MIMOFormer. Our code is available at https://github.com/IBM/multiple-input-multiple-output-nets.

Learning with rejection is an important framework that can refrain from making predictions to avoid critical mispredictions by balancing between prediction and rejection. Previous studies on cost-based rejection only focused on the classification setting, which cannot handle the continuous and infinite target space in the regression setting. In this paper, we investigate a novel regression problem called regression with cost-based rejection, where the model can reject to make predictions on some examples given certain rejection costs. To solve this problem, we first formulate the expected risk for this problem and then derive the Bayes optimal solution, which shows that the optimal model should reject to make predictions on the examples whose variance is larger than the rejection cost when the mean squared error is used as the evaluation metric. Furthermore, we propose to train the model by a surrogate loss function that considers rejection as binary classification and we provide conditions for the model consistency, which implies that the Bayes optimal solution can be recovered by our proposed surrogate loss. Extensive experiments demonstrate the effectiveness of our proposed method.

We present a method to formulate algorithm discovery as program search, and apply it to discover optimization algorithms for deep neural network training. We leverage efficient search techniques to explore an infinite and sparse program space. To bridge the large generalization gap between proxy and target tasks, we also introduce program selection and simplification strategies. Our method discovers a simple and effective optimization algorithm, $\textbf{Lion}$ ($\textit{Evo$\textbf{L}$ved S$\textbf{i}$gn M$\textbf{o}$me$\textbf{n}$tum}$). It is more memory-efficient than Adam as it only keeps track of the momentum. Different from adaptive optimizers, its update has the same magnitude for each parameter calculated through the sign operation. We compare Lion with widely used optimizers, such as Adam and Adafactor, for training a variety of models on different tasks. On image classification, Lion boosts the accuracy of ViT by up to 2\% on ImageNet and saves up to 5x the pre-training compute on JFT. On vision-language contrastive learning, we achieve 88.3\% $\textit{zero-shot}$ and 91.1\% $\textit{fine-tuning}$ accuracy on ImageNet, surpassing the previous best results by 2\% and 0.1\%, respectively. On diffusion models, Lion outperforms Adam by achieving a better FID score and reducing the training compute by up to 2.3x. For autoregressive, masked language modeling, and fine-tuning, Lion exhibits a similar or better performance compared to Adam. Our analysis of Lion reveals that its performance gain grows with the training batch size. It also requires a smaller learning rate than Adam due to the larger norm of the update produced by the sign function. Additionally, we examine the limitations of Lion and identify scenarios where its improvements are small or not statistically significant.

Closed-form differential equations, including partial differential equations and higher-order ordinary differential equations, are one of the most important tools used by scientists to model and better understand natural phenomena. Discovering these equations directly from data is challenging because it requires modeling relationships between various derivatives that are not observed in the data (equation-data mismatch) and it involves searching across a huge space of possible equations. Current approaches make strong assumptions about the form of the equation and thus fail to discover many well-known phenomena. Moreover, many of them resolve the equation-data mismatch by estimating the derivatives, which makes them inadequate for noisy and infrequent observations. To this end, we propose D-CIPHER, which is robust to measurement artifacts and can uncover a new and very general class of differential equations. We further design a novel optimization procedure, CoLLie, to help D-CIPHER search through this class efficiently. Finally, we demonstrate empirically that it can discover many well-known equations that are beyond the capabilities of current methods.

To the best of our knowledge, there are no methods today for training differentially private regression models on sparse input data. To remedy this, we adapt the Frank-Wolfe algorithm for $L_1$ penalized linear regression to be aware of sparse inputs and to use them effectively. In doing so, we reduce the training time of the algorithm from $\mathcal{O}( T D S + T N S)$ to $\mathcal{O}(N S + T \sqrt{D} \log{D} + T S^2)$, where $T$ is the number of iterations and a sparsity rate $S$ of a dataset with $N$ rows and $D$ features. Our results demonstrate that this procedure can reduce runtime by a factor of up to $2,200\times$, depending on the value of the privacy parameter $\epsilon$ and the sparsity of the dataset.

Fr\'echet regression has emerged as a promising approach for regression analysis involving non-Euclidean response variables. However, its practical applicability has been hindered by its reliance on ideal scenarios with abundant and noiseless covariate data. In this paper, we present a novel estimation method that tackles these limitations by leveraging the low-rank structure inherent in the covariate matrix. Our proposed framework combines the concepts of global Fr\'echet regression and principal component regression, aiming to improve the efficiency and accuracy of the regression estimator. By incorporating the low-rank structure, our method enables more effective modeling and estimation, particularly in high-dimensional and errors-in-variables regression settings. We provide a theoretical analysis of the proposed estimator's large-sample properties, including a comprehensive rate analysis of bias, variance, and additional variations due to measurement errors. Furthermore, our numerical experiments provide empirical evidence that supports the theoretical findings, demonstrating the superior performance of our approach. Overall, this work introduces a promising framework for regression analysis of non-Euclidean variables, effectively addressing the challenges associated with limited and noisy covariate data, with potential applications in diverse fields.

We explore the minimax optimal error associated with a demographic parity-constrained regression problem within the context of a linear model. Our proposed model encompasses a broader range of discriminatory bias sources compared to the model presented by Chzhen and Schreuder. Our analysis reveals that the minimax optimal error for the demographic parity-constrained regression problem under our model is characterized by $\Theta(\frac{dM}{n})$, where $n$ denotes the sample size, $d$ represents the dimensionality, and $M$ signifies the number of demographic groups arising from sensitive attributes. Moreover, we demonstrate that the minimax error increases in conjunction with a larger bias present in the model.

Contextual batched bandit (CBB) is a setting where a batch of rewards is observed from the environment at the end of each episode, but the rewards of the non-executed actions are unobserved, resulting in partial-information feedback. Existing approaches for CBB often ignore the rewards of the non-executed actions, leading to underutilization of feedback information. In this paper, we propose an efficient approach called Sketched Policy Updating with Imputed Rewards (SPUIR) that completes the unobserved rewards using sketching, which approximates the full-information feedbacks. We formulate reward imputation as an imputation regularized ridge regression problem that captures the feedback mechanisms of both executed and non-executed actions. To reduce time complexity, we solve the regression problem using randomized sketching. We prove that our approach achieves an instantaneous regret with controllable bias and smaller variance than approaches without reward imputation. Furthermore, our approach enjoys a sublinear regret bound against the optimal policy. We also present two extensions, a rate-scheduled version and a version for nonlinear rewards, making our approach more practical. Experimental results show that SPUIR outperforms state-of-the-art baselines on synthetic, public benchmark, and real-world datasets.

We study pointwise estimation and uncertainty quantification for a sparse variational Gaussian process method with eigenvector inducing variables. For a rescaled Brownian motion prior, we derive theoretical guarantees and limitations for the frequentist size and coverage of pointwise credible sets. For sufficiently many inducing variables, we precisely characterize the asymptotic frequentist coverage, deducing when credible sets from this variational method are conservative and when overconfident/misleading. We numerically illustrate the applicability of our results and discuss connections with other common Gaussian process priors.

We consider an important problem in scientific discovery, namely identifying sparse governing equations for nonlinear dynamical systems. This involves solving sparse ridge regression problems to provable optimality in order to determine which terms drive the underlying dynamics. We propose a fast algorithm, OKRidge, for sparse ridge regression, using a novel lower bound calculation involving, first, a saddle point formulation, and from there, either solving (i) a linear system or (ii) using an ADMM-based approach, where the proximal operators can be efficiently evaluated by solving another linear system and an isotonic regression problem. We also propose a method to warm-start our solver, which leverages a beam search. Experimentally, our methods attain provable optimality with run times that are orders of magnitude faster than those of the existing MIP formulations solved by the commercial solver Gurobi.

We study the performance of empirical risk minimization on the $p$-norm linear regression problem for $p \in (1, \infty)$. We show that, in the realizable case, under no moment assumptions, and up to a distribution-dependent constant, $O(d)$ samples are enough to exactly recover the target. Otherwise, for $p \in [2, \infty)$, and under weak moment assumptions on the target and the covariates, we prove a high probability excess risk bound on the empirical risk minimizer whose leading term matches, up to a constant that depends only on $p$, the asymptotically exact rate. We extend this result to the case $p \in (1, 2)$ under mild assumptions that guarantee the existence of the Hessian of the risk at its minimizer.