Large language model (LLM) tokenizers act as structured compressors: by mapping text to discrete token sequences, they determine token count (and thus compute and context usage) and the statistical structure seen by downstream models. Despite their central role in LLM pipelines, the link between tokenization, compression efficiency and induced structure is not well understood. We empirically demonstrate that tokenizer training scale redistributes entropy: as training data grows, the token stream becomes more diverse in aggregate (higher unigram entropy) yet markedly more predictable in-context (lower higher-order conditional entropies), indicating that tokenization absorbs substantial short-range regularity although these gains degrade under train-test domain mismatch. To ground these observations, we first benchmark i) pretrained GPT-family tokenizers as black-box compressors across various domains, and ii) learned tokenizers across configurations spanning vocabulary size, training scale, and domain. Next, we study tokenization as a transform for universal compression and introduce a compression-aware BPE variant. Finally, we adopt a channel lens and introduce capacity-utilization metrics to analyze tokenizer behaviour and outline implications for downstream modeling. Put together, our results expose various trade-offs between compression, induced structure, and robustness under domain shift, and motivate principled, compression-aware tokenizer design.

We present ConvexECG, an explainable and resource-efficient method for reconstructing six-lead electrocardiograms (ECG) from single-lead data, aimed at advancing personalized and continuous cardiac monitoring. ConvexECG leverages a convex reformulation of a two-layer ReLU neural network, enabling the potential for efficient training and deployment in resource constrained environments, while also having deterministic and explainable behavior. Using data from 25 patients, we demonstrate that ConvexECG achieves accuracy comparable to larger neural networks while significantly reducing computational overhead, highlighting its potential for real-time, low-resource monitoring applications.

Previous research has shown that the principal singular vectors of a pre-trained model’s weight matrices capture critical knowledge. In contrast, those associated with small singular values may contain noise or less reliable information. As a result, the LoRA-based parameter-efficient fine-tuning (PEFT) approach, which does not constrain the use of the spectral space, may not be effective for tasks that demand high representation capacity. In this study, we enhance existing PEFT techniques by incorporating the spectral information of pre-trained weight matrices into the fine-tuning process. We investigate spectral adaptation strategies with a particular focus on the additive adjustment of top singular vectors. This is accomplished by applying singular value decomposition (SVD) to the pre-trained weight matrices and restricting the fine-tuning within the top spectral space. Extensive speaker verification experiments on VoxCeleb1 and CN-Celeb1 demonstrate enhanced tuning performance with the proposed approach. Code is released at https://github.com/lizhepolyu/SpectralFT.

Transformer architectures are the backbone of the modern AI revolution. However, these models simply stack the same blocks in dozens of layers and process information sequentially from one block to another. In this paper, we propose to challenge this and introduce adaptive computations for LLM-like setups, which allow the final layer to attend to all of the intermediate layers as it deems fit through the attention mechanism, thereby introducing computational attention shortcuts. These shortcuts can thus make the architecture depth and context adaptive. We showcase four different datasets, namely acoustic tokens, natural language, and symbolic music, and we achieve superior performance for GPT-like architecture. We give evidence via attention maps that the models learn complex dependencies across layers that are adaptive in context and depth depending on the input tokens.

We present Subtractive Training1, a simple and novel method for synthesizing individual musical instrument stems given other instruments as context. This method pairs a dataset of complete music mixes with 1) a variant of the dataset lacking a specific stem, and 2) LLM-generated instructions describing how the missing stem should be reintroduced. We then fine-tune a pre-trained text-to-audio diffusion model to generate the missing instrument stem, guided by both the existing stems and the text instruction. Our results demonstrate Subtractive Training’s efficacy in creating authentic instrument stems that seamlessly blend with the existing tracks. We also show that we can use the text instruction to control the generation of the inserted stem in terms of rhythm, dynamics, and genre, allowing us to modify the style of a single instrument in a full song while keeping the remaining instruments the same. Lastly, we extend this technique to MIDI formats, successfully generating compatible bass, drum, and guitar parts for incomplete arrangements.

We discuss several aspects of the loss landscape of regularized neural networks: the structure of stationary points, connectivity of optimal solutions, path with non-increasing loss to arbitrary global optimum, and the nonuniqueness of optimal solutions, by casting the problem into an equivalent convex problem and considering its dual. Starting from two-layer neural networks with scalar output, we first characterize the solution set of the convex problem using its dual and further characterize all stationary points. With the characterization, we show that the topology of the global optima goes through a phase transition as the width of the network changes, and construct counterexamples where the problem may have a continuum of optimal solutions. Finally, we show that the solution set characterization and connectivity results can be extended to different architectures, including two layer vector-valued neural networks and parallel three-layer neural networks.

Motivated by recent advances in serverless cloud computing, in particular the ``function as a service'' (FaaS) model, we consider the problem of minimizing a convex function in a massively parallel fashion, where communication between workers is limited. Focusing on the case of a twice-differentiable objective subject to an L2 penalty, we propose a scheme where the central node (server) effectively runs a Newton method, offloading its high per-iteration cost---stemming from the need to invert the Hessian---to the workers. In our solution, workers produce independently coarse but low-bias estimates of the inverse Hessian, using an adaptive sketching scheme. The server then averages the descent directions produced by the workers, yielding a good approximation for the exact Newton step. The main component of our adaptive sketching scheme is a low-complexity procedure for selecting the sketching dimension, an issue that was left largely unaddressed in the existing literature on Hessian sketching for distributed optimization. Our solution is based on ideas from asymptotic random matrix theory, specifically the Marchenko-Pastur law. For Gaussian sketching matrices, we derive non asymptotic guarantees for our algorithm which do not depend on the condition number of the Hessian nor a priori require the sketching dimension to be proportional to the dimension, as is often the case in asymptotic random matrix theory. Lastly, when the objective is self-concordant, we provide convergence guarantees for the approximate Newton's method with noisy Hessians, which may be of independent interest beyond the setting considered in this paper.

Volume parameterizations abound in recent literature, encompassing methods from classic voxel grids to implicit neural representations. While implicit representations offer impressive capacity and improved memory efficiency compared to voxel grids, they traditionally require training through nonconvex optimization, which can be slow and sensitive to initialization and hyperparameters. We introduce GA-Planes, a novel family of implicit neural volume representations inspired by Geometric Algebra that can be trained using convex optimization, addressing the limitations of nonconvex methods. GA-Planes models generalize many existing representations including any combination of features stored in tensor basis elements followed by a neural feature decoder, and can be adapted to convex or nonconvex training as needed for various inverse problems. In the 2D setting, we prove GA-Planes models are equivalent to a low-rank plus low-resolution matrix factorization that outperforms the classic low-rank plus sparse decomposition for fitting a natural image. In 3D, GA-Planes models exhibit competitive expressiveness, model size, and optimizability across tasks such as radiance field reconstruction, 3D segmentation, and video segmentation.

Active learning methods aim to improve sample complexity in machine learning. In this work, we investigate an active learning scheme via a novel gradient-free cutting-plane training method for ReLU networks of arbitrary depth and develop a convergence theory. We demonstrate, for the first time, that cutting-plane algorithms, traditionally used in linear models, can be extended to deep neural networks despite their nonconvexity and nonlinear decision boundaries. Moreover, this training method induces the first deep active learning scheme known to achieve convergence guarantees, revealing a geometric contraction rate of the feasible set. We exemplify the effectiveness of our proposed active learning method against popular deep active learning baselines via both synthetic data experiments and sentimental classification task on real datasets.

Due to the non-convex nature of training Deep Neural Network (DNN) models, their effectiveness relies on the use of non-convex optimization heuristics. Traditional methods for training DNNs often require costly empirical methods to produce successful models and do not have a clear theoretical foundation. In this study, we examine the use of convex optimization theory and sparse recovery models to refine the training process of neural networks and provide a better interpretation of their optimal weights. We focus on training two-layer neural networks with piecewise linear activations and demonstrate that they can be formulated as a finite-dimensional convex program. These programs include a regularization term that promotes sparsity, which constitutes a variant of group Lasso. We first utilize semi-infinite programming theory to prove strong duality for finite width neural networks and then we express these architectures equivalently as high dimensional convex sparse recovery models. Remarkably, the worst-case complexity to solve the convex program is polynomial in the number of samples and number of neurons when the rank of the data matrix is bounded, which is the case in convolutional networks. To extend our method to training data of arbitrary rank, we develop a novel polynomial-time approximation scheme based on zonotope subsampling that comes with a guaranteed approximation ratio. We also show that all the stationary points of the nonconvex training objective can be characterized as the global optimum of a subsampled convex program. Our convex models can be trained using standard convex solvers without resorting to heuristics or extensive hyper-parameter tuning unlike non-convex methods. Due to the convexity, optimizer hyperparameters such as initialization, batch sizes, and step size schedules have no effect on the final model. Through extensive numerical experiments, we show that convex models can outperform traditional non-convex methods and are not sensitive to optimizer hyperparameters. The code for our experiments is available athttps://github.com/pilancilab/convex_nn.

The practice of deep learning has shown that neural networks generalize remarkably well even with an extreme number of learned parameters. This appears to contradict traditional statistical wisdom, in which a trade-off between model complexity and fit to the data is essential. We aim to address this discrepancy by adopting a convex optimization and sparse recovery perspective. We consider the training and generalization properties of two-layer ReLU networks with standard weight decay regularization. Under certain regularity assumptions on the data, we show that ReLU networks with an arbitrary number of parameters learn only simple models that explain the data. This is analogous to the recovery of the sparsest linear model in compressed sensing. For ReLU networks and their variants with skip connections or normalization layers, we present isometry conditions that ensure the exact recovery of planted neurons. For randomly generated data, we show the existence of a phase transition in recovering planted neural network models, which is easy to describe: whenever the ratio between the number of samples and the dimension exceeds a numerical threshold, the recovery succeeds with high probability; otherwise, it fails with high probability. Surprisingly, ReLU networks learn simple and sparse models that generalize well even when the labels are noisy. The phase transition phenomenon is confirmed through numerical experiments.

Data centers increasingly consume large amounts of power. In this paper, we develop an efficient and scalable power management scheme, based on optimizing the tradeoff between a processor's speed and its response/delay time to process jobs. Processors/servers are lightly coordinated by a system manager that sets an internal power “price” signal to discourage excessive power usage. The stability of the power management scheme depends on the power pricing function, and a rule of thumb is given for pricing functions that guarantee stability. The analysis is consistent with simulations, which demonstrate the behavior of the scheme. It can be implemented in a decentralized manner across servers/processors, making it scalable to large data centers.

It has been demonstrated that the training problem for a variety of (non) linear two-layer neural networks (such as two-layer perceptrons, convolutional networks, and self-attention) can be posed as equivalent convex optimization problems, with an induced regularizer which encourages low rank. However, this regularizer becomes prohibitively expensive to compute at moderate scales, impeding training convex neural networks. To this end, we propose applying the Burer-Monteiro factorization to convex neural networks, which for the first time enables a Burer-Monteiro perspective on neural networks with non-linearities. This factorization leads to an equivalent yet computationally tractable non-convex alternative with no spurious local minima. We develop a novel relative optimality bound of stationary points of the Burer-Monteiro factorization, providing verifiable conditions under which any stationary point is a global optimum. Further, for the first time, we show that linear self-attention with sufficiently many heads has no spurious local minima. Our experiments validate the novel relative optimality bound and the utility of the Burer-Monteiro factorization for scaling convex neural networks.

In this paper, we study the optimality gap between two-layer ReLU networks regularized with weight decay and their convex relaxations. We show that when the training data is random, the relative optimality gap between the original problem and its relaxation can be bounded by a factor of O(√log n), where n is the number of training samples. A simple application leads to a tractable polynomial-time algorithm that is guaranteed to solve the original non-convex problem up to a logarithmic factor. Moreover, under mild assumptions, we show that local gradient methods converge to a point with low training loss with high probability. Our result is an exponential improvement compared to existing results and sheds new light on understanding why local gradient methods work well.

Low-Rank Adaptation (LoRA) emerges as a popular parameter-efficient fine-tuning (PEFT) method, which proposes to freeze pretrained model weights and update an additive low-rank trainable matrix. In this work, we study the enhancement of LoRA training by introducing an $r\times r$ preconditioner in each gradient step where $r$ is the LoRA rank. We theoretically verify that the proposed preconditioner stabilizes feature learning with LoRA under infinite-width NN setting. Empirically, the implementation of this new preconditioner requires a small change to existing optimizer code and creates virtually minuscule storage and runtime overhead. Our experimental results with both large language models and text-to-image diffusion models show that with this new preconditioner, the convergence and reliability of SGD and AdamW can be significantly enhanced. Moreover, the training process becomes much more robust to hyperparameter choices such as learning rate. The new preconditioner can be derived from a novel Riemannian metric in low-rank matrix field.

The softmax function is ubiquitous in machine learning and optimization applications. Computing the full softmax evaluation of a matrix-vector product can be computationally expensive in high-dimensional settings. In many applications, however, it is sufficient to calculate only the top few outputs of the softmax function. In this work, we present an algorithm, dubbed AdaptiveSoftmax, that adaptively computes the top k softmax values more efficiently than the full softmax computation, with probabilistic guarantees. We demonstrate the sample efficiency improvements afforded by AdaptiveSoftmax on real and synthetic data to corroborate our theoretical results. AdaptiveSoftmax yields >10x gain over full softmax computation on most datasets, yielding up to 30x improvement for Mistral7B evaluated on the Wikitext dataset. The adaptive method we propose for estimating the partition function (the softmax denominator) is of independent interest and can be used in other applications such as kernel density estimation.

We introduce the CRONOS algorithm for convex optimization of two-layer neural networks. CRONOS is the first algorithm capable of scaling to high-dimensional datasets such as ImageNet, which are ubiquitous in modern deep learning. This significantly improves upon prior work, which has been restricted to downsampled versions of MNIST and CIFAR-10. Taking CRONOS as a primitive, we then develop a new algorithm called CRONOS-AM, which combines CRONOS with alternating minimization, to obtain an algorithm capable of training multi-layer networks with arbitrary architectures. Our theoretical analysis proves that CRONOS converges to the global minimum of the convex reformulation under mild assumptions. In addition, we validate the efficacy of CRONOS and CRONOS-AM through extensive large-scale numerical experiments with GPU acceleration in JAX. Our results show that CRONOS-AM can obtain comparable or better validation accuracy than predominant tuned deep learning optimizers on vision and language tasks with benchmark datasets such as ImageNet and IMDb. To the best of our knowledge, CRONOS is the first algorithm which utilizes the convex reformulation to enhance performance on large-scale learning tasks.

The prohibitive sizes of Large Language Models (LLMs) today make it difficult to deploy them on memory-constrained edge devices. This work introduces $\rm CALDERA$ -- a new post-training LLM compression algorithm that harnesses the inherent low-rank structure of a weight matrix $\mathbf{W}$ by approximating it via a low-rank, low-precision decomposition as $\mathbf{W} \approx \mathbf{Q} + \mathbf{L}\mathbf{R}$. Here, $\mathbf{L}$ and $\mathbf{R}$ are low rank factors, and the entries of $\mathbf{Q}$, $\mathbf{L}$ and $\mathbf{R}$ are quantized. The model is compressed by substituting each layer with its $\mathbf{Q} + \mathbf{L}\mathbf{R}$ decomposition, and the zero-shot performance of the compressed model is evaluated. Additionally, $\mathbf{L}$ and $\mathbf{R}$ are readily amenable to low-rank adaptation, consequently enhancing the zero-shot performance. $\rm CALDERA$ obtains this decomposition by formulating it as an optimization problem $\min_{\mathbf{Q},\mathbf{L},\mathbf{R}}\lVert(\mathbf{Q} + \mathbf{L}\mathbf{R} - \mathbf{W})\mathbf{X}^\top\rVert_{\rm F}^2$, where $\mathbf{X}$ is the calibration data, and $\mathbf{Q}, \mathbf{L}, \mathbf{R}$ are constrained to be representable using low-precision formats. Theoretical upper bounds on the approximation error of $\rm CALDERA$ are established using a rank-constrained regression framework, and the tradeoff between compression ratio and model performance is studied by analyzing the impact of target rank and quantization bit budget. Results illustrate that compressing LlaMa-$2$ $7$B/$13$B/$70$B and LlaMa-$3$ $8$B models obtained using $\rm CALDERA$ outperforms existing post-training LLM compression techniques in the regime of less than $2.5$ bits per parameter.

Recent developments in Parameter-Efficient Fine-Tuning (PEFT) methods for pretrained deep neural networks have captured widespread interest. In this work, we study the enhancement of current PEFT methods by incorporating the spectral information of pretrained weight matrices into the fine-tuning procedure. We investigate two spectral adaptation mechanisms, namely additive tuning and orthogonal rotation of the top singular vectors, both are done via first carrying out Singular Value Decomposition (SVD) of pretrained weights and then fine-tuning the top spectral space. We provide a theoretical analysis of spectral fine-tuning and show that our approach improves the rank capacity of low-rank adapters given a fixed trainable parameter budget. We show through extensive experiments that the proposed fine-tuning model enables better parameter efficiency and tuning performance as well as benefits multi-adapter fusion. The source code will be open-sourced for reproducibility.

Gradient coding is a method for mitigating straggling servers in a centralized computing network that uses erasure-coding techniques to distributively carry out first-order optimization methods. Randomized numerical linear algebra uses randomization to develop improved algorithms for large-scale linear algebra computations. In this paper, we propose a method for distributed optimization that combines gradient coding and randomized numerical linear algebra. The proposed method uses a randomized$\ell _{2}$-subspace embedding and a gradient coding technique to distribute blocks of data to the computational nodes of a centralized network, and at each iteration the central server only requires a small number of computations to obtain the steepest descent update. The novelty of our approach is that the data is replicated according to importance scores, called block leverage scores, in contrast to most gradient coding approaches that uniformly replicate the data blocks. Furthermore, we do not require a decoding step at each iteration, avoiding a bottleneck in previous gradient coding schemes. We show that our approach results in a valid$\ell _{2}$-subspace embedding, and that our resulting approximation converges to the optimal solution.

We consider training multi-layer neural networks with polynomial and ReLU activation functions. We develop exact convex optimization formulations for three-layer and deeper architectures. Our formulations are based on semidefinite lifting and recent results on the hidden convexity of two-layer ReLU networks. We show that certain deep neural network architectures can be trained to global optimality in polynomial time by solving equivalent convex optimization problems.

The large memory footprint of high dimensional models require quantization to a lower precision for deployment on resource constrained edge devices. With this motivation, we consider the problems of learning a (i) linear regressor, and a (ii) linear classifier from a given training dataset, and quantizing the learned model parameters subject to a pre-specified bit-budget. The error metric is the prediction risk of the quantized model, and our proposed randomized embedding-based quantization methods attain near-optimal error while being computationally efficient. We provide fundamental bounds on the bit-budget constrained minimax risk that, together with our proposed algorithms, characterize the minimum threshold budget required to achieve a risk comparable to the unquantized setting. We also show the efficacy of our strategy by quantizing a two-layer ReLU neural network for non-linear regression. Numerical simulations show the improved performance of our proposed scheme as well as its closeness to the lower bound.

We develop an analytical framework to characterize the set of optimal ReLU neural networks by reformulating the non-convex training problem as a convex program. We show that the global optima of the convex parameterization are given by a polyhedral set and then extend this characterization to the optimal set of the non-convex training objective. Since all stationary points of the ReLU training problem can be represented as optima of sub-sampled convex programs, our work provide a general expression for all critical points of the non-convex objective. We then leverage our results to provide an optimal pruning algorithm for computing minimal networks, establish conditions for the regularization path of ReLU networks to be continuous, and develop sensitivity results for minimal ReLU networks.

In this work, we address the problem of Hessian inversion bias in distributed second-order optimization algorithms. We introduce a novel shrinkage-based estimator for the resolvent of gram matrices which is asymptotically unbiased, and characterize its non-asymptotic convergence rate in the isotropic case. We apply this estimator to bias correction of Newton steps in distributed second-order optimization algorithms, as well as randomized sketching based methods. We examine the bias present in the naive averaging-based distributed Newton's method using analytical expressions and contrast it with our proposed biasfree approach. Our approach leads to significant improvements in convergence rate compared to standard baselines and recent proposals, as shown through experiments on both real and synthetic datasets.

Recently, theoretical analyses of deep neural networks have broadly focused on two directions: 1) Providing insight into neural network training by SGD in the limit of infinite hidden-layer width and infinitesimally small learning rate (also known as gradient flow) via the Neural Tangent Kernel (NTK), and 2) Globally optimizing the regularized training objective via cone-constrained convex reformulations of ReLU networks. The latter research direction also yielded an alternative formulation of the ReLU network, called a gated ReLU network, that is globally optimizable via efficient unconstrained convex programs. In this work, we interpret the convex program for this gated ReLU network as a Multiple Kernel Learning (MKL) model with a weighted data masking feature map and establish a connection to the NTK. Specifically, we show that for a particular choice of mask weights that do not depend on the learning targets, this kernel is equivalent to the NTK of the gated ReLU network on the training data. A consequence of this lack of dependence on the targets is that the NTK cannot perform better than the optimal MKL kernel on the training set. By using iterative reweighting, we improve the weights induced by the NTK to obtain the optimal MKL kernel which is equivalent to the solution of the exact convex reformulation of the gated ReLU network. We also provide several numerical simulations corroborating our theory. Additionally, we provide an analysis of the prediction error of the resulting optimal kernel via consistency results for the group lasso.

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.

Matrices are exceptionally useful in various fields of study as they provide a convenient framework to organize and manipulate data in a structured manner. However, modern matrices can involve billions of elements, making their storage and processing quite demanding in terms of computational resources and memory usage. Although prohibitively large, such matrices are often approximately low rank. We propose an algorithm that exploits this structure to obtain a low rank decomposition of any matrix $\mathbf{A}$ as $\mathbf{A} \approx \mathbf{L}\mathbf{R}$, where $\mathbf{L}$ and $\mathbf{R}$ are the low rank factors. The total number of elements in $\mathbf{L}$ and $\mathbf{R}$ can be significantly less than that in $\mathbf{A}$. Furthermore, the entries of $\mathbf{L}$ and $\mathbf{R}$ are quantized to low precision formats -- compressing $\mathbf{A}$ by giving us a low rank and low precision factorization. Our algorithm first computes an approximate basis of the range space of $\mathbf{A}$ by randomly sketching its columns, followed by a quantization of the vectors constituting this basis. It then computes approximate projections of the columns of $\mathbf{A}$ onto this quantized basis. We derive upper bounds on the approximation error of our algorithm, and analyze the impact of target rank and quantization bit-budget. The tradeoff between compression ratio and approximation accuracy allows for flexibility in choosing these parameters based on specific application requirements. We empirically demonstrate the efficacy of our algorithm in image compression, nearest neighbor classification of image and text embeddings, and compressing the layers of LlaMa-$7$b. Our results illustrate that we can achieve compression ratios as aggressive as one bit per matrix coordinate, all while surpassing or maintaining the performance of traditional compression techniques.

We consider distributed optimization methods for problems where forming the Hessian is computationally challenging and communication is a significant bottleneck. We leverage randomized sketches for reducing the problem dimensions as well as preserving privacy and improving straggler resilience in asynchronous distributed systems. We derive novel approximation guarantees for classical sketching methods and establish tight concentration results that serve as both upper and lower bounds on the error. We then extend our analysis to the accuracy of parameter averaging for distributed sketches. Furthermore, we develop unbiased parameter averaging methods for randomized second order optimization in regularized problems that employ sketching of the Hessian. Existing works do not take the bias of the estimators into consideration, which limits their application to massively parallel computation. We provide closed-form formulas for regularization parameters and step sizes that provably minimize the bias for sketched Newton directions. Additionally, we demonstrate the implications of our theoretical findings via large scale experiments on a serverless cloud computing platform.

We study the problem of estimating the value of a known smooth function f at an unknown point $\mu \in \mathbb{R}^n$, where each component $\mu_i$ can be sampled via a noisy oracle. Sampling more frequently components of $\mu$ corresponding to directions of the function with larger directional derivatives is more sample-efficient. However, as $\mu$ is unknown, the optimal sampling frequencies are also unknown. We design an instance-adaptive algorithm that learns to sample according to the importance of each coordinate, and with probability at least $1-\delta$ returns an $\epsilon$ accurate estimate of $f(\mu)$. We generalize our algorithm to adapt to heteroskedastic noise, and prove asymptotic optimality when f is linear. We corroborate our theoretical results with numerical experiments, showing the dramatic gains afforded by adaptivity.

Fisher’s Linear Discriminant Analysis (FLDA) is a statistical analysis method that linearly embeds data points to a lower dimensional space to maximize a discrimination criterion such that the variance between classes is maximized while the variance within classes is minimized. We introduce a natural extension of FLDA that employs neural networks, called Neural Fisher Discriminant Analysis (NFDA). This method finds the optimal two-layer neural network that embeds data points to optimize the same discrimination criterion. We use tools from convex optimization to transform the optimal neural network embedding problem into a convex problem. The resulting problem is easy to interpret and solve to global optimality. We evaluate the method’s performance on synthetic and real datasets.

We develop fast algorithms and robust software for convex optimization of two-layer neural networks with ReLU activation functions. Our work leverages a convex re-formulation of the standard weight-decay penalized training problem as a set of group-l1-regularized data-local models, where locality is enforced by polyhedral cone constraints. In the special case of zero-regularization, we show that this problem is exactly equivalent to unconstrained optimization of a convex "gated ReLU" network. For problems with non-zero regularization, we show that convex gated ReLU models obtain data-dependent approximation bounds for the ReLU training problem. To optimize the convex re-formulations, we develop an accelerated proximal gradient method and a practical augmented Lagrangian solver. We show that these approaches are faster than standard training heuristics for the non-convex problem, such as SGD, and outperform commercial interior-point solvers. Experimentally, we verify our theoretical results, explore the group-l1 regularization path, and scale convex optimization for neural networks to image classification on MNIST and CIFAR-10.

Vision transformers using self-attention or its proposed alternatives have demonstrated promising results in many image related tasks. However, the underpinning inductive bias of attention is not well understood. To address this issue, this paper analyzes attention through the lens of convex duality. For the non-linear dot-product self-attention, and alternative mechanisms such as MLP-mixer and Fourier Neural Operator (FNO), we derive equivalent finite-dimensional convex problems that are interpretable and solvable to global optimality. The convex programs lead to block nuclear-norm regularization that promotes low rank in the latent feature and token dimensions. In particular, we show how self-attention networks implicitly clusters the tokens, based on their latent similarity. We conduct experiments for transferring a pre-trained transformer backbone for CIFAR-100 classification by fine-tuning a variety of convex attention heads. The results indicate the merits of the bias induced by attention compared with the existing MLP or linear heads.

In this work, we propose a method for speeding up linear regression distributively, while ensuring security. We leverage randomized sketching techniques, and improve straggler resilience in asynchronous systems. Specifically, we apply a random orthonormal matrix and then subsample in blocks, to simultaneously secure the information and reduce the dimension of the regression problem. In our setup, the transformation corresponds to an encoded encryption in an approximate gradient coding scheme, and the subsampling corresponds to the responses of the non-straggling workers; in a centralized coded computing network. We focus on the special case of the Subsampled Randomized Hadamard Transform, which we generalize to block sampling; and discuss how it can be used to secure the data.

The ever-increasing need for real-time 3D perception for autonomy has resulted in development of a new generation of high-resolution perception sensors paired with powerful edge processors and advanced perception algorithms. However, processing of the large data volumes generated by such high-resolution sensors still imposes a great challenge for resource-limited edge devices and systems. In this work, we visit this problem in the context of pedestrian gait analysis using high-resolution mm-Wave sensors with application in autonomous driving. We propose reducing the total computational load of a resource-limited perception system by dynamically controlling the mm-Wave sensor’s resolution on the fly. We provide results on three dynamic sensor adaptation approaches including an end-to-end Deep Q-Learning approach that can overcome the resource and performance limitations inherent to conventional static methods.

We propose novel randomized optimization methods for high-dimensional convex problems based on restrictions of variables to random subspaces. We consider oblivious and data-adaptive subspaces and study their approximation properties via convex duality and Fenchel conjugates. A suitable adaptive subspace can be generated by sampling a correlated random matrix whose second order statistics mirror the input data. We illustrate that the adaptive strategy can significantly outperform the standard oblivious sampling method, which is widely used in the recent literature. We show that the relative error of the randomized approximations can be tightly characterized in terms of the spectrum of the data matrix and Gaussian width of the dual tangent cone at optimum. We develop lower bounds for both optimization and statistical error measures based on concentration of measure and Fano’s inequality. We then present the consequences of our theory with data matrices of varying spectral decay profiles. Experimental results show that the proposed approach enables significant speed ups in a wide variety of machine learning and optimization problems including logistic regression, kernel classification with random convolution layers and shallow neural networks with rectified linear units.

We introduce an error resilient distributed computing method based on an extension of the channel polarization phenomenon to distributed algorithms. The method leverages an algorithmic split operation that transforms two identical compute nodes into slow and fast workers, which parallels the channel split operation in Polar Codes. This operation preserves the average runtime, analogous to the conservation of Shannon capacity in channel polarization. By leveraging a recursive construction in a similar spirit to the Fast Fourier Transform, the method synthesizes virtual compute nodes with dispersed return time distributions, which we call computational polarization. We show that the runtime distributions form a functional martingale process, identify their limiting distributions in closed-form expressions together with non-asymptotic convergence rates, and prove strong convergence results in Banach spaces. We provide an information-theoretic lower bound on the overall runtime of any coded computation method and show that the computational polarization approach asymptotically achieves the optimal runtime for computing linear functions. An important advantage is the near linear time decoding procedure, which is significantly faster than Maximum Distance Separable codes.

One of the most common operations in signal processing is matrix multiplication. However, it presents a major computational bottleneck when the matrix dimension is high, as can occur for large data size or feature dimension. Two different approaches to overcoming this bottleneck are: 1) low rank approximation of the matrix product; and 2) distributed computation. We propose a scheme that combines these two approaches. To enable distributed low rank approximation, we generalize the approximate matrix CR-multiplication to accommodate weighted block sampling, and we introduce a weighted coded matrix multiplication method. This results in novel approximate weighted CR coded matrix multiplication schemes, which achieve improved performance for distributed matrix multiplication and are robust to stragglers.

Three features are crucial for sequential forecasting and generation models: tractability, expressiveness, and theoretical backing. While neural autoregressive models are relatively tractable and offer powerful predictive and generative capabilities, they often have complex optimization landscapes, and their theoretical properties are not well understood. To address these issues, we present convex formulations of autoregressive models with one hidden layer. Specifically, we prove an exact equivalence between these models and constrained, regularized logistic regression by using semi-infinite duality to embed the data matrix onto a higher dimensional space and introducing inequality constraints. To make this formulation tractable, we approximate the constraints using a hinge loss or drop them altogether. Furthermore, we demonstrate faster training and competitive performance of these implementations compared to their neural network counterparts on a variety of data sets. Consequently, we introduce techniques to derive tractable, expressive, and theoretically-interpretable models that are nearly equivalent to neural autoregressive models.

Neural networks (NNs) have been extremely successful across many tasks in machine learning. Quantization of NN weights has become an important topic due to its impact on their energy efficiency, inference time and deployment on hardware. Although post-training quantization is well-studied, training optimal quantized NNs involves combinatorial non-convex optimization problems which appear intractable. In this work, we introduce a convex optimization strategy to train quantized NNs with polynomial activations. Our method leverages hidden convexity in two-layer neural networks from the recent literature, semidefinite lifting, and Grothendieck’s identity. Surprisingly, we show that certain quantized NN problems can be solved to global optimality provably in polynomial time in all relevant parameters via tight semidefinite relaxations. We present numerical examples to illustrate the effectiveness of our method.

Understanding the fundamental mechanism behind the success of deep neural networks is one of the key challenges in the modern machine learning literature. Despite numerous attempts, a solid theoretical analysis is yet to be developed. In this paper, we develop a novel unified framework to reveal a hidden regularization mechanism through the lens of convex optimization. We first show that the training of multiple three-layer ReLU sub-networks with weight decay regularization can be equivalently cast as a convex optimization problem in a higher dimensional space, where sparsity is enforced via a group $\ell_1$-norm regularization. Consequently, ReLU networks can be interpreted as high dimensional feature selection methods. More importantly, we then prove that the equivalent convex problem can be globally optimized by a standard convex optimization solver with a polynomial-time complexity with respect to the number of samples and data dimension when the width of the network is fixed. Finally, we numerically validate our theoretical results via experiments involving both synthetic and real datasets.

We study regularized deep neural networks (DNNs) and introduce a convex analytic framework to characterize the structure of the hidden layers. We show that a set of optimal hidden layer weights for a norm regularized DNN training problem can be explicitly found as the extreme points of a convex set. For the special case of deep linear networks, we prove that each optimal weight matrix aligns with the previous layers via duality. More importantly, we apply the same characterization to deep ReLU networks with whitened data and prove the same weight alignment holds. As a corollary, we also prove that norm regularized deep ReLU networks yield spline interpolation for one-dimensional datasets which was previously known only for two-layer networks. Furthermore, we provide closed-form solutions for the optimal layer weights when data is rank-one or whitened. The same analysis also applies to architectures with batch normalization even for arbitrary data. Therefore, we obtain a complete explanation for a recent empirical observation termed Neural Collapse where class means collapse to the vertices of a simplex equiangular tight frame.

We propose a randomized algorithm with quadratic convergence rate for convex optimization problems with a self-concordant, composite, strongly convex objective function. Our method is based on performing an approximate Newton step using a random projection of the Hessian. Our first contribution is to show that, at each iteration, the embedding dimension (or sketch size) can be as small as the effective dimension of the Hessian matrix. Leveraging this novel fundamental result, we design an algorithm with a sketch size proportional to the effective dimension and which exhibits a quadratic rate of convergence. This result dramatically improves on the classical linear-quadratic convergence rates of state-of-the-art sub-sampled Newton methods. However, in most practical cases, the effective dimension is not known beforehand, and this raises the question of how to pick a sketch size as small as the effective dimension while preserving a quadratic convergence rate. Our second and main contribution is thus to propose an adaptive sketch size algorithm with quadratic convergence rate and which does not require prior knowledge or estimation of the effective dimension: at each iteration, it starts with a small sketch size, and increases it until quadratic progress is achieved. Importantly, we show that the embedding dimension remains proportional to the effective dimension throughout the entire path and that our method achieves state-of-the-art computational complexity for solving convex optimization programs with a strongly convex component. We discuss and illustrate applications to linear and quadratic programming, as well as logistic regression and other generalized linear models.

In second-order optimization, a potential bottleneck can be computing the Hessian matrix of the optimized function at every iteration. Randomized sketching has emerged as a powerful technique for constructing estimates of the Hessian which can be used to perform approximate Newton steps. This involves multiplication by a random sketching matrix, which introduces a trade-off between the computational cost of sketching and the convergence rate of the optimization. A theoretically desirable but practically much too expensive choice is to use a dense Gaussian sketching matrix, which produces unbiased estimates of the exact Newton step and offers strong problem-independent convergence guarantees. We show that the Gaussian matrix can be drastically sparsified, substantially reducing the computational cost, without affecting its convergence properties in any way. This approach, called Newton-LESS, is based on a recently introduced sketching technique: LEverage Score Sparsified (LESS) embeddings. We prove that Newton-LESS enjoys nearly the same problem-independent local convergence rate as Gaussian embeddings for a large class of functions. In particular, this leads to a new state-of-the-art convergence result for an iterative least squares solver. Finally, we substantially extend LESS embeddings to include uniformly sparsified random sign matrices which can be implemented efficiently and perform well in numerical experiments.

We develop a convex analytic approach to analyze finite width two-layer ReLU networks. We first prove that an optimal solution to the regularized training problem can be characterized as extreme points of a convex set, where simple solutions are encouraged via its convex geometrical properties. We then leverage this characterization to show that an optimal set of parameters yield linear spline interpolation for regression problems involving one dimensional or rank-one data. We also characterize the classification decision regions in terms of a kernel matrix and minimum $\ell_1$-norm solutions. This is in contrast to Neural Tangent Kernel which is unable to explain predictions of finite width networks. Our convex geometric characterization also provides intuitive explanations of hidden neurons as auto-encoders. In higher dimensions, we show that the training problem can be cast as a finite dimensional convex problem with infinitely many constraints. Then, we apply certain convex relaxations and introduce a cutting-plane algorithm to globally optimize the network. We further analyze the exactness of the relaxations to provide conditions for the convergence to a global optimum. Our analysis also shows that optimal network parameters can be also characterized as interpretable closed-form formulas in some practically relevant special cases.

We develop a convex analytic framework for ReLU neural networks which elucidates the inner workings of hidden neurons and their function space characteristics. We show that rectified linear units in neural networks act as convex regularizers, where simple solutions are encouraged via extreme points of a certain convex set. For one dimensional regression and classification, we prove that finite two-layer ReLU networks with norm regularization yield linear spline interpolation. In the more general higher dimensional case, we show that the training problem for two-layer networks can be cast as a convex optimization problem with infinitely many constraints. We then provide a family of convex relaxations to approximate the solution, and a cutting-plane algorithm to improve the relaxations. We derive conditions for the exactness of the relaxations and provide simple closed form formulas for the optimal neural network weights in certain cases. Our results show that the hidden neurons of a ReLU network can be interpreted as convex autoencoders of the input layer. We also establish a connection to $\ell_0$-$\ell_1$ equivalence for neural networks analogous to the minimal cardinality solutions in compressed sensing. Extensive experimental results show that the proposed approach yields interpretable and accurate models.

A major hurdle in machine learning is scalability to massive datasets. Approaches to overcome this hurdle include compression of the data matrix and distributing the computations. Leverage score sampling provides a compressed approximation of a data matrix using an importance weighted subset. Gradient coding has been recently proposed in distributed optimization to compute the gradient using multiple unreliable worker nodes. By designing coding matrices, gradient coded computations can be made resilient to stragglers, which are nodes in a distributed network that degrade system performance. We present a novel weighted leverage score approach, that achieves improved performance for distributed gradient coding by utilizing an importance sampling.

We provide an exact analysis of a class of randomized algorithms for solving overdetermined least-squares problems. We consider first-order methods, where the gradients are pre-conditioned by an approximation of the Hessian, based on a subspace embedding of the data matrix. This class of algorithms encompasses several randomized methods among the fastest solvers for least-squares problems. We focus on two classical embeddings, namely, Gaussian projections and subsampled randomized Hadamard transforms (SRHT). Our key technical innovation is the derivation of the limiting spectral density of SRHT embeddings. Leveraging this novel result, we derive the family of normalized orthogonal polynomials of the SRHT density and we find the optimal pre-conditioned first-order method along with its rate of convergence. Our analysis of Gaussian embeddings proceeds similarly, and leverages classical random matrix theory results. In particular, we show that for a given sketch size, SRHT embeddings exhibits a faster rate of convergence than Gaussian embeddings. Then, we propose a new algorithm by optimizing the computational complexity over the choice of the sketching dimension. To our knowledge, our resulting algorithm yields the best known complexity for solving least-squares problems with no condition number dependence.

We develop exact representations of training two-layer neural networks with rectified linear units (ReLUs) in terms of a single convex program with number of variables polynomial in the number of training samples and the number of hidden neurons. Our theory utilizes semi-infinite duality and minimum norm regularization. We show that ReLU networks trained with standard weight decay are equivalent to block $\ell_1$ penalized convex models. Moreover, we show that certain standard convolutional linear networks are equivalent semi-definite programs which can be simplified to $\ell_1$ regularized linear models in a polynomial sized discrete Fourier feature space

In distributed second order optimization, a standard strategy is to average many local estimates, each of which is based on a small sketch or batch of the data. However, the local estimates on each machine are typically biased, relative to the full solution on all of the data, and this can limit the effectiveness of averaging. Here, we introduce a new technique for debiasing the local estimates, which leads to both theoretical and empirical improvements in the convergence rate of distributed second order methods. Our technique has two novel components: (1) modifying standard sketching techniques to obtain what we call a surrogate sketch; and (2) carefully scaling the global regularization parameter for local computations. Our surrogate sketches are based on determinantal point processes, a family of distributions for which the bias of an estimate of the inverse Hessian can be computed exactly. Based on this computation, we show that when the objective being minimized is $l_2$-regularized with parameter $\lambda$ and individual machines are each given a sketch of size $m$, then to eliminate the bias, local estimates should be computed using a shrunk regularization parameter given by $\lambda^\prime=\lambda(1-\frac{d_\lambda}{m})$, where $d_\lambda$ is the $\lambda$-effective dimension of the Hessian (or, for quadratic problems, the data matrix).

Random projections or sketching are widely used in many algorithmic and learning contexts. Here we study the performance of iterative Hessian sketch for least-squares problems. By leveraging and extending recent results from random matrix theory on the limiting spectrum of matrices randomly projected with the subsampled randomized Hadamard transform, and truncated Haar matrices, we can study and compare the resulting algorithms to a level of precision that has not been possible before. Our technical contributions include a novel formula for the second moment of the inverse of projected matrices. We also find simple closed-form expressions for asymptotically optimal step-sizes and convergence rates. These show that the convergence rate for Haar and randomized Hadamard matrices are identical, and asymptotically improve upon Gaussian random projections. These techniques may be applied to other algorithms that employ randomized dimension reduction.

We propose a new randomized algorithm for solving L2-regularized least-squares problems based on sketching. We consider two of the most popular random embeddings, namely, Gaussian embeddings and the Subsampled Randomized Hadamard Transform (SRHT). While current randomized solvers for least-squares optimization prescribe an embedding dimension at least greater than the data dimension, we show that the embedding dimension can be reduced to the effective dimension of the optimization problem, and still preserve high-probability convergence guarantees. In this regard, we derive sharp matrix deviation inequalities over ellipsoids for both Gaussian and SRHT embeddings. Specifically, we improve on the constant of a classical Gaussian concentration bound whereas, for SRHT embeddings, our deviation inequality involves a novel technical approach. Leveraging these bounds, we are able to design a practical and adaptive algorithm which does not require to know the effective dimension beforehand. Our method starts with an initial embedding dimension equal to 1 and, over iterations, increases the embedding dimension up to the effective one at most. Hence, our algorithm improves the state-of-the-art computational complexity for solving regularized least-squares problems. Further, we show numerically that it outperforms standard iterative solvers such as the conjugate gradient method and its pre-conditioned version on several standard machine learning datasets.

We propose convex relaxations for convolutional neural nets with one hidden layer where the output weights are fixed. For convex activation functions such as rectified linear units, the relaxations are convex second order cone programs which can be solved very efficiently. We prove that the relaxation recovers the global minimum under a planted model assumption, given sufficiently many training samples from a Gaussian distribution. We also identify a phase transition phenomenon in recovering the global minimum for the relaxation.

We propose a novel randomized linear least squares solver which is an improvement of Iterative Hessian Sketch and randomized preconditioning. In the proposed Momentum-IHS technique (M-IHS), Heavy Ball Method is used to accelerate the convergence of iterations. It is shown that for any full rank data matrix, rate of convergence depends on the ratio between the feature size and the sketch size. Unlike the Conjugate Gradient technique, the rate of convergence is unaffected by either the condition number or the eigenvalue spectrum of the data matrix. As demonstrated over many examples, the proposed M-IHS provides compatible performance with the state of the art randomized preconditioning methods such as LSRN or Blendenpik and yet, it provides a completely different perspective in the area of iterative solvers which can pave the way for future developments.

We propose a new randomized optimization method for high-dimensional problems which can be seen as a generalization of coordinate descent to random subspaces. We show that an adaptive sampling strategy for the random subspace significantly outperforms the oblivious sampling method, which is the common choice in the recent literature. The adaptive subspace can be efficiently generated by a correlated random matrix ensemble whose statistics mimic the input data. We prove that the improvement in the relative error of the solution can be tightly characterized in terms of the spectrum of the data matrix, and provide probabilistic upper-bounds. We then illustrate the consequences of our theory with data matrices of different spectral decay. Extensive experimental results show that the proposed approach offers significant speed ups in machine learning problems including logistic regression, kernel classification with random convolution layers and shallow neural networks with rectified linear units. Our analysis is based on convex analysis and Fenchel duality, and establishes connections to sketching and randomized matrix decompositions.

We study randomized sketching methods for approximately solving least-squares problem with a general convex constraint. The quality of a least-squares approximation can be assessed in different ways: either in terms of the value of the quadratic objective function (cost approximation), or in terms of some distance measure between the approximate minimizer and the true minimizer (solution approximation). Focusing on the latter criterion, our first main result provides a general lower bound on any randomized method that sketches both the data matrix and vector in a least-squares problem; as a surprising consequence, the most widely used least-squares sketch is sub-optimal for solution approximation. We then present a new method known as the iterative Hessian sketch, and show that it can be used to obtain approximations to the original least-squares problem using a projection dimension proportional to the statistical complexity of the least-squares minimizer, and a logarithmic number of iterations. We illustrate our general theory with simulations for both unconstrained and constrained versions of least-squares, including $\ell_1$-regularization and nuclear norm constraints. We also numerically demonstrate the practicality of our approach in a real face expression classification experiment.

Random projection (RP) is a classical technique for reducing storage and computational costs. We analyze RP-based approximations of convex programs, in which the original optimization problem is approximated by solving a lower dimensional problem. Such dimensionality reduction is essential in computation-limited settings, since the complexity of general convex programming can be quite high (e.g., cubic for quadratic programs, and substantially higher for semidefinite programs). In addition to computational savings, RP is also useful for reducing memory usage, and has useful properties for privacy-preserving optimization. We prove that the approximation ratio of this procedure can be bounded in terms of the geometry of the constraint set. For a broad class of RPs, including those based on various sub-Gaussian distributions as well as randomized Hadamard and Fourier transforms, the data matrix defining the cost function can be projected to a dimension proportional to the squared Gaussian width of the tangent cone of the constraint set at the original solution. This effective dimension of the convex program is often substantially smaller than the original dimension. We illustrate consequences of our theory for various cases, including unconstrained and L1-constrained least squares, support vector machines, low-rank matrix estimation, and discuss implications for privacy-preserving optimization, as well as connections with denoising and compressed sensing.

A novel expectation maximization based matching pursuit (EMMP) algorithm is presented. The method uses the measurements as the incomplete data and obtain the complete data which corresponds to the sparse solution using an iterative EM based framework. In standard greedy methods such as matching pursuit or orthogonal matching pursuit a selected atom can not be changed during the course of the algorithm even if the signal doesn't have a support on that atom. The proposed EMMP algorithm is also flexible in that sense. The results show that the proposed method has lower reconstruction errors compared to other greedy algorithms using the same conditions.

We consider the problem of cardinality penalized optimization of a convex function over the probability simplex with additional convex constraints. It's well-known that the classical L1 regularizer fails to promote sparsity on the probability simplex since L1 norm on the probability simplex is trivially constant. We propose a direct relaxation of the minimum cardinality problem and show that it can be efficiently solved using convex programming. As a first application we consider recovering a sparse probability measure given moment constraints, in which our formulation becomes linear programming, hence can be solved very efficiently. A sufficient condition for exact recovery of the minimum cardinality solution is derived for arbitrary affine constraints. We then develop a penalized version for the noisy setting which can be solved using second order cone programs. The proposed method outperforms known heuristics based on L1 norm. As a second application we consider convex clustering using a sparse Gaussian mixture and compare our results with the well known soft k-means algorithm.

We show that the exact recovery of sparse perturbations on the coefficient matrix in overdetermined Least Squares problems is possible for a large class of perturbation structures. The well established theory of Compressed Sensing enables us to prove that if the perturbation structure is sufficiently incoherent, then exact or stable recovery can be achieved using linear programming. We derive sufficiency conditions for both exact and stable recovery using known results of ℓ0/ℓ1equivalence. However the problem turns out to be more complicated than the usual setting used in various sparse reconstruction problems. We propose and solve an optimization criterion and its convex relaxation to recover the perturbation and the solution to the Least Squares problem simultaneously. Then we demonstrate with numerical examples that the proposed method is able to recover the perturbation and the unknown exactly with high probability. The performance of the proposed technique is compared in blind identification of sparse multipath channels.

In many signal processing applications the core problem reduces to a linear system of equations. Coefficient matrix uncertainties create a significant challenge in obtaining reliable solutions. In this paper, we present a novel formulation for solving a system of noise contaminated linear equations while preserving the structure of the coefficient matrix. The proposed method has advantages over the known Structured Total Least Squares (STLS) techniques in utilizing additional information about the uncertainties and robustness in ill-posed problems. Numerical comparisons are given to illustrate these advantages in two applications: signal restoration problem with an uncertain model and frequency estimation of multiple sinusoids embedded in white noise.