We focus on the task of learning a single index model $\sigma(w^\star \cdot x)$ with respect to the isotropic Gaussian distribution in $d$ dimensions. Prior work has shown that the sample complexity of learning $w^\star$ is governed by the information exponent $k^\star$ of the link function $\sigma$, which is defined as the index of the first nonzero Hermite coefficient of $\sigma$. Ben Arous et al. (2021) showed that $n \gtrsim d^{k^\star-1}$ samples suffice for learning $w^\star$ and that this is tight for online SGD. However, the CSQ lower bound for gradient based methods only shows that $n \gtrsim d^{k^\star/2}$ samples are necessary. In this work, we close the gap between the upper and lower bounds by showing that online SGD on a smoothed loss learns $w^\star$ with $n \gtrsim d^{k^\star/2}$ samples. We also draw connections to statistical analyses of tensor PCA and to the implicit regularization effects of minibatch SGD on empirical losses.

Neural sequence models, especially transformers, exhibit a remarkable capacity for in-context learning. They can construct new predictors from sequences of labeled examples $(x, f(x))$ presented in the input without further parameter updates. We investigate the hypothesis that transformer-based in-context learners implement standard learning algorithms implicitly, by encoding smaller models in their activations, and updating these implicit models as new examples appear in the context. Using linear regression as a prototypical problem, we offer three sources of evidence for this hypothesis. First, we prove by construction that transformers can implement learning algorithms for linear models based on gradient descent and closed-form ridge regression. Second, we show that trained in-context learners closely match the predictors computed by gradient descent, ridge regression, and exact least-squares regression, transitioning between different predictors as transformer depth and dataset noise vary, and converging to Bayesian estimators for large widths and depths. Third, we present preliminary evidence that in-context learners share algorithmic features with these predictors: learners' late layers non-linearly encode weight vectors and moment matrices. These results suggest that in-context learning is understandable in algorithmic terms, and that (at least in the linear case) learners may rediscover standard estimation algorithms. Code and reference implementations are released at https://github.com/ekinakyurek/google-research/blob/master/incontext.

We show the convergence of Wasserstein inverse reinforcement learning for multi-objective optimizations with the projective subgradient method by formulating an inverse problem of the multi-objective optimization problem. In addition, we prove convergence of inverse reinforcement learning (maximum entropy inverse reinforcement learning, guided cost learning) with gradient descent and the projective subgradient method.

Large pre-trained language models (PLMs) have garnered significant attention for their versatility and potential for solving a wide spectrum of natural language processing (NLP) tasks. However, the cost of running these PLMs may be prohibitive. Furthermore, PLMs may not be open-sourced due to commercial considerations and potential risks of misuse, such as GPT-3. The parameters and gradients of PLMs are unavailable in this scenario. To solve the issue, black-box tuning has been proposed, which utilizes derivative-free optimization (DFO), instead of gradient descent, for training task-specific continuous prompts. However, these gradient-free methods still exhibit a significant gap compared to gradient-based methods. In this paper, we introduce gradient descent into black-box tuning scenario through knowledge distillation. Furthermore, we propose a novel method GDFO, which integrates gradient descent and derivative-free optimization to optimize task-specific continuous prompts in a harmonized manner. Experimental results show that GDFO can achieve significant performance gains over previous state-of-the-art methods.

We present a novel approach for black-box VI that bypasses the difficulties of stochastic gradient ascent, including the task of selecting step-sizes. Our approach involves using a sequence of sample average approximation (SAA) problems. SAA approximates the solution of stochastic optimization problems by transforming them into deterministic ones. We use quasi-Newton methods and line search to solve each deterministic optimization problem and present a heuristic policy to automate hyperparameter selection. Our experiments show that our method simplifies the VI problem and achieves faster performance than existing methods.

Many machine learning applications and tasks rely on the stochastic gradient descent (SGD) algorithm and its variants. Effective step length selection is crucial for the success of these algorithms, which has motivated the development of algorithms such as ADAM or AdaGrad. In this paper, we propose a novel algorithm for adaptive step length selection in the classical SGD framework, which can be readily adapted to other stochastic algorithms. Our proposed algorithm is inspired by traditional nonlinear optimization techniques and is supported by analytical findings. We show that under reasonable conditions, the algorithm produces step lengths in line with well-established theoretical requirements, and generates iterates that converge to a stationary neighborhood of a solution in expectation. We test the proposed algorithm on logistic regressions and deep neural networks and demonstrate that the algorithm can generate step lengths comparable to the best step length obtained from manual tuning.

Achieving communication efficiency in decentralized machine learning has been attracting significant attention, with communication compression recognized as an effective technique in algorithm design. This paper takes a first step to understand the role of gradient clipping, a popular strategy in practice, in decentralized nonconvex optimization with communication compression. We propose PORTER, which considers two variants of gradient clipping added before or after taking a mini-batch of stochastic gradients, where the former variant PORTER-DP allows local differential privacy analysis with additional Gaussian perturbation, and the latter variant PORTER-GC helps to stabilize training. We develop a novel analysis framework that establishes their convergence guarantees without assuming the stringent bounded gradient assumption. To the best of our knowledge, our work provides the first convergence analysis for decentralized nonconvex optimization with gradient clipping and communication compression, highlighting the trade-offs between convergence rate, compression ratio, network connectivity, and privacy.

The Noisy-SGD algorithm is widely used for privately training machine learning models. Traditional privacy analyses of this algorithm assume that the internal state is publicly revealed, resulting in privacy loss bounds that increase indefinitely with the number of iterations. However, recent findings have shown that if the internal state remains hidden, then the privacy loss might remain bounded. Nevertheless, this remarkable result heavily relies on the assumption of (strong) convexity of the loss function. It remains an important open problem to further relax this condition while proving similar convergent upper bounds on the privacy loss. In this work, we address this problem for DP-SGD, a popular variant of Noisy-SGD that incorporates gradient clipping to limit the impact of individual samples on the training process. Our findings demonstrate that the privacy loss of projected DP-SGD converges exponentially fast, without requiring convexity or smoothness assumptions on the loss function. In addition, we analyze the privacy loss of regularized (unprojected) DP-SGD. To obtain these results, we directly analyze the hockey-stick divergence between coupled stochastic processes by relying on non-linear data processing inequalities.

In Federated Learning (FL) client devices connected over the internet collaboratively train a machine learning model without sharing their private data with a central server or with other clients. The seminal Federated Averaging (FedAvg) algorithm trains a single global model by performing rounds of local training on clients followed by model averaging. FedAvg can improve the communication-efficiency of training by performing more steps of Stochastic Gradient Descent (SGD) on clients in each round. However, client data in real-world FL is highly heterogeneous, which has been extensively shown to slow model convergence and harm final performance when $K > 1$ steps of SGD are performed on clients per round. In this work we propose decaying $K$ as training progresses, which can jointly improve the final performance of the FL model whilst reducing the wall-clock time and the total computational cost of training compared to using a fixed $K$. We analyse the convergence of FedAvg with decaying $K$ for strongly-convex objectives, providing novel insights into the convergence properties, and derive three theoretically-motivated decay schedules for $K$. We then perform thorough experiments on four benchmark FL datasets (FEMNIST, CIFAR100, Sentiment140, Shakespeare) to show the real-world benefit of our approaches in terms of real-world convergence time, computational cost, and generalisation performance.

Recent research in neural networks and machine learning suggests that using many more parameters than strictly required by the initial complexity of a regression problem can result in more accurate or faster-converging models -- contrary to classical statistical belief. This phenomenon, sometimes known as ``benign overfitting'', raises questions regarding in what other ways might overparameterization affect the properties of a learning problem. In this work, we investigate the effects of overfitting on the robustness of gradient-descent training when subject to uncertainty on the gradient estimation. This uncertainty arises naturally if the gradient is estimated from noisy data or directly measured. Our object of study is a linear neural network with a single, arbitrarily wide, hidden layer and an arbitrary number of inputs and outputs. In this paper we solve the problem for the case where the input and output of our neural-network are one-dimensional, deriving sufficient conditions for robustness of our system based on necessary and sufficient conditions for convergence in the undisturbed case. We then show that the general overparametrized formulation introduces a set of spurious equilibria which lay outside the set where the loss function is minimized, and discuss directions of future work that might extend our current results for more general formulations.