Understanding the fundamental principles behind the success of deep neural networks is one of the most important open questions in the current literature.

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Understanding the fundamental principles behind the success of deep neural networks is one of the most important open questions in the current literature. To this end, we study the training problem of deep neural networks and introduce an analytic approach to unveil hidden convexity in the optimization landscape. We consider a deep parallel ReLU network architecture, which also includes standard deep networks and ResNets as its special cases. We then show that pathwise regularized training problems can be represented as an exact convex optimization problem. We further prove that the equivalent convex problem is regularized via a group sparsity inducing norm. Thus, a path regularized parallel ReLU network can be viewed as a parsimonious convex model in high dimensions. More importantly, since the original training problem may not be trainable in polynomial-time, we propose an approximate algorithm with a fully polynomial-time complexity in all data dimensions. Then, we prove strong global optimality guarantees for this algorithm. We also provide experiments corroborating our theory.

Current methods for training Binarized Neural Networks (BNNs) heavily rely on the heuristic straight-through estimator (STE), which crucially enables the application of SGD-based optimizers to the combinatorial training problem. Although the STE heuristics and their variants have led to significant improvements in BNN performance, their theoretical underpinnings remain unclear and relatively understudied. In this paper, we propose a theoretically motivated optimization framework for BNN training based on Gaussian variational inference. In its simplest form, our approach yields a non-convex linear programming formulation whose variables and associated gradients motivate the use of latent weights and STE gradients. More importantly, our framework allows us to formulate semidefinite programming (SDP) relaxations to the BNN training task. Such formulations are able to explicitly models pairwise correlations between weights during training, leading to a more accurate optimization characterization of the training problem. As the size of such formulations grows quadratically in the number of weights, quickly becoming intractable for large networks, we apply the Burer-Monteiro approach and only optimize over linear-size low-rank SDP solutions. Our empirical evaluation on CIFAR-10, CIFAR-100, Tiny-ImageNet and ImageNet datasets shows our method consistently outperforming all state-of-the-art algorithms for training BNNs.

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Despite their widespread use, training deep Transformers can be unstable. Layer normalization, a standard component, improves training stability, but its placement has often been ad-hoc. In this paper, we conduct a principled study on the forward (hidden states) and backward (gradient) stability of Transformers under different layer normalization placements. Our theory provides key insights into the training dynamics: whether training drives Transformers toward regular solutions or pathological behaviors. For forward stability, we derive explicit bounds on the growth of hidden states in trained Transformers. For backward stability, we analyze how layer normalization affects the backprop-agation of gradients, thereby explaining the training dynamics of each layer normalization placement. Our analysis also guides the scaling of residual steps in Transformer blocks, where appropriate choices can further improve stability and performance. Our numerical results corroborate our theoretical findings. Beyond these results, our framework provides a principled way to sanity-check the stability of Transformers under new architectural modifications, offering guidance for future designs.

Understanding the fundamental principles behind the success of deep neural networks is one of the most important open questions in the current literature.

--Learning the structure of Bayesian networks (BNs) from data is challenging, especially for datasets involving a large number of variables. The recently proposed divide-and-conquer (D&D) strategies present a promising approach for learning large BNs. However, they still face a main issue of unstable learning accuracy across subproblems. In this work, we introduce the idea of employing structure learning ensemble (SLE), which combines multiple BN structure learning algorithms, to consistently achieve high learning accuracy. We further propose an automatic approach called Auto-SLE for learning near-optimal SLEs, addressing the challenge of manually designing high-quality SLEs. The learned SLE is then integrated into a D&D method. Extensive experiments firmly show the superiority of our method over D&D methods with single BN structure learning algorithm in learning large BNs, achieving accuracy improvement usually by 30% 225% on datasets involving 10,000 variables. These results indicate the significant potential of employing (automatic learning of) SLEs for scalable BN structure learning. Learning the structure of Bayesian networks (BNs) [1] from data has attracted much research interest, due to its wide applications in machine learning, statistical modeling, and causal inference [2]-[4].

The non-convexity of the artificial neural network (ANN) training landscape brings inherent optimization difficulties. While the traditional back-propagation stochastic gradient descent (SGD) algorithm and its variants are effective in certain cases, they can become stuck at spurious local minima and are sensitive to initializations and hyperparameters. Recent work has shown that the training of an ANN with ReLU activations can be reformulated as a convex program, bringing hope to globally optimizing interpretable ANNs. However, naively solving the convex training formulation has an exponential complexity, and even an approximation heuristic requires cubic time. In this work, we characterize the quality of this approximation and develop two efficient algorithms that train ANNs with global convergence guarantees. The first algorithm is based on the alternating direction method of multiplier (ADMM). It solves both the exact convex formulation and the approximate counterpart. Linear global convergence is achieved, and the initial several iterations often yield a solution with high prediction accuracy. When solving the approximate formulation, the per-iteration time complexity is quadratic. The second algorithm, based on the "sampled convex programs" theory, solves unconstrained convex formulations and converges to an approximately globally optimal classifier. The non-convexity of the ANN training landscape exacerbates when adversarial training is considered. We apply the robust convex optimization theory to convex training and develop convex formulations that train ANNs robust to adversarial inputs. Our analysis explicitly focuses on one-hidden-layer fully connected ANNs, but can extend to more sophisticated architectures.

Current methods for training Binarized Neural Networks (BNNs) heavily rely on the heuristic straight-through estimator (STE), which crucially enables the application of SGD-based optimizers to the combinatorial training problem. Although the STE heuristics and their variants have led to significant improvements in BNN performance, their theoretical underpinnings remain unclear and relatively understudied. In this paper, we propose a theoretically motivated optimization framework for BNN training based on Gaussian variational inference. In its simplest form, our approach yields a non-convex linear programming formulation whose variables and associated gradients motivate the use of latent weights and STE gradients. More importantly, our framework allows us to formulate semidefinite programming (SDP) relaxations to the BNN training task.