We study policy-regret minimization problem in dynamically evolving environments, modeled as Markov games between a learner and a strategic, adaptive opponent. We propose a general algorithmic framework that achieves the optimal $\mathcal{O}(\sqrt{T})$ policy regret for a wide class of large-scale problems characterized by an Eluder-type condition–extending beyond the tabular settings of previous work. Importantly, our framework uncovers a simpler yet powerful algorithmic approach for handling reactive adversaries, demonstrating that leveraging opponent learning in such settings is key to attaining the optimal $\mathcal{O}(\sqrt{T})$ policy regret.

Despite the remarkable success of large foundation models across a range of tasks, they remain susceptible to security threats such as backdoor attacks. By injecting poisoned data containing specific triggers during training, adversaries can manipulate model predictions in a targeted manner. While prior work has focused on empirically designing and evaluating such attacks, a rigorous theoretical understanding of when and why they succeed is lacking. In this work, we analyze backdoor attacks that exploit the token selection process within attention mechanisms--a core component of transformer-based architectures. We show that single-head self-attention transformers trained via gradient descent can interpolate poisoned training data. Moreover, we prove that when the backdoor triggers are sufficiently strong but not overly dominant, attackers can successfully manipulate model predictions. Our analysis characterizes how adversaries manipulate token selection to alter outputs and identifies the theoretical conditions under which these attacks succeed. We validate our findings through experiments on synthetic datasets.

Curriculum learning has emerged as an effective strategy to enhance the training efficiency and generalization of machine learning models. However, its theoretical underpinnings remain relatively underexplored. In this work, we develop a theoretical framework for curriculum learning based on biased regularized empirical risk minimization (RERM), identifying conditions under which curriculum learning provably improves generalization. We introduce a sufficient condition that characterizes a "good" curriculum and analyze a multi-task curriculum framework, where solving a sequence of convex tasks can facilitate better generalization. We also demonstrate how these theoretical insights translate to practical benefits when using stochastic gradient descent (SGD) as an optimization method. Beyond convex settings, we explore the utility of curriculum learning for non-convex tasks. Empirical evaluations on synthetic datasets and MNIST validate our theoretical findings and highlight the practical efficacy of curriculum-based training.

We study private empirical risk minimization (ERM) problem for losses satisfying the $(\gamma,\kappa)$-Kurdyka-{Ł}ojasiewicz (KL) condition, that is, the empirical loss $F$ satisfies $F(w)-\min_{w}F(w) \leq \gamma^\kappa \|\nabla F(w)\|^\kappa$. The Polyak-{Ł}ojasiewicz (PL) condition is a special case of this condition when $\kappa=2$. Specifically, we study this problem under the constraint of $\rho$ zero-concentrated differential privacy (zCDP). When $\kappa\in[1,2]$ and the loss function is Lipschitz and smooth over a sufficiently large region, we provide a new algorithm based on variance reduced gradient descent that achieves the rate $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrt{\rho}}\big)^\kappa\big)$ on the excess empirical risk, where $n$ is the dataset size and $d$ is the dimension. We further show that this rate is nearly optimal. When $\kappa \geq 2$ and the loss is instead Lipschitz and weakly convex, we show it is possible to achieve the rate $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrt{\rho}}\big)^\kappa\big)$ with a private implementation of the proximal point method. When the KL parameters are unknown, we provide a novel modification and analysis of the noisy gradient descent algorithm and show that this algorithm achieves a rate of $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrt{\rho}}\big)^{\frac{2\kappa}{4-\kappa}}\big)$ adaptively, which is nearly optimal when $\kappa = 2$. We further show that, without assuming the KL condition, the same gradient descent algorithm can achieve fast convergence to a stationary point when the gradient stays sufficiently large during the run of the algorithm. Specifically, we show that this algorithm can approximate stationary points of Lipschitz, smooth (and possibly nonconvex) objectives with rate as fast as $\tilde{O}\big(\frac{\sqrt{d}}{n\sqrt{\rho}}\big)$ and never worse than $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrt{\rho}}\big)^{1/2}\big)$. The latter rate matches the best known rate for methods that do not rely on variance reduction.

We study the statistical complexity of offline decision-making with function approximation, establishing (near) minimax-optimal rates for stochastic contextual bandits and Markov decision processes. The performance limits are captured by the pseudo-dimension of the (value) function class and a new characterization of the behavior policy that *strictly* subsumes all the previous notions of data coverage in the offline decision-making literature. In addition, we seek to understand the benefits of using offline data in online decision-making and show nearly minimax-optimal rates in a wide range of regimes.

In this work, we explore Hypothesis Transfer Learning (HTL) under adversarial attacks. In this setting, a learner has access to a training dataset of size $n$ from an underlying distribution $\mathcal{D}$ and a set of auxiliary hypotheses. These auxiliary hypotheses, which can be viewed as prior information originating either from expert knowledge or as pre-trained foundation models, are employed as an initialization for the learning process. Our goal is to develop an adversarially robust model for $\mathcal{D}$. We begin by examining an adversarial variant of the regularized empirical risk minimization learning rule that we term A-RERM. Assuming a non-negative smooth loss function with a strongly convex regularizer, we establish a bound on the robust generalization error of the hypothesis returned by A-RERM in terms of the robust empirical loss and the quality of the initialization. If the initialization is good, i.e., there exists a weighted combination of auxiliary hypotheses with a small robust population loss, the bound exhibits a fast rate of $\mathcal{O}(1/n)$. Otherwise, we get the standard rate of $\mathcal{O}(1/\sqrt{n})$. Additionally, we provide a bound on the robust excess risk which is similar in nature, albeit with a slightly worse rate. We also consider solving the problem using a practical variant, namely proximal stochastic adversarial training, and present a bound that depends on the initialization. This bound has the same dependence on the sample size as the ARERM bound, except for an additional term that depends on the size of the adversarial perturbation.

Benign overfitting is the phenomenon wherein none of the predictors in the hypothesis class can achieve perfect accuracy (i.e., non-realizable or noisy setting), but a model that interpolates the training data still achieves good generalization. A series of recent works aim to understand this phenomenon for regression and classification tasks using linear predictors as well as two-layer neural networks. In this paper, we study such a benign overfitting phenomenon in an adversarial setting. We show that under a distributional assumption, interpolating neural networks found using adversarial training generalize well despite inference-time attacks. Specifically, we provide convergence and generalization guarantees for adversarial training of two-layer networks (with smooth as well as non-smooth activation functions) showing that under moderate $\ell_2$ norm perturbation budget, the trained model has near-zero robust training loss and near-optimal robust generalization error. We support our theoretical findings with an empirical study on synthetic and real-world data.

We study offline multitask representation learning in reinforcement learning (RL), where a learner is provided with an offline dataset from different tasks that share a common representation and is asked to learn the shared representation. We theoretically investigate offline multitask low-rank RL, and propose a new algorithm called MORL for offline multitask representation learning. Furthermore, we examine downstream RL in reward-free, offline and online scenarios, where a new task is introduced to the agent that shares the same representation as the upstream offline tasks. Our theoretical results demonstrate the benefits of using the learned representation from the upstream offline task instead of directly learning the representation of the low-rank model.

We study learning in a dynamically evolving environment modeled as a Markov game between a learner and a strategic opponent that can adapt to the learner's strategies. While most existing works in Markov games focus on external regret as the learning objective, external regret becomes inadequate when the adversaries are adaptive. In this work, we focus on \emph{policy regret} -- a counterfactual notion that aims to compete with the return that would have been attained if the learner had followed the best fixed sequence of policy, in hindsight. We show that if the opponent has unbounded memory or if it is non-stationary, then sample-efficient learning is not possible. For memory-bounded and stationary, we show that learning is still statistically hard if the set of feasible strategies for the learner is exponentially large. To guarantee learnability, we introduce a new notion of \emph{consistent} adaptive adversaries, wherein, the adversary responds similarly to similar strategies of the learner. We provide algorithms that achieve $\sqrt{T}$ policy regret against memory-bounded, stationary, and consistent adversaries.

We study the limits and capability of public-data assisted differentially private (PA-DP) algorithms. Specifically, we focus on the problem of stochastic convex optimization (SCO) with either labeled or unlabeled public data. For complete/labeled public data, we show that any $(\epsilon,\delta)$-PA-DP has excess risk $\tilde{\Omega}\big(\min(\frac{1}{\sqrt{n_{\text{pub}}}},\frac{1}{\sqrt{n}}+\frac{\sqrt{d}}{n\epsilon} ) \big)$, where $d$ is the dimension, ${n_{\text{pub}}}$ is the number of public samples, ${n_{\text{priv}}}$ is the number of private samples, and $n={n_{\text{pub}}}+{n_{\text{priv}}}$. These lower bounds are established via our new lower bounds for PA-DP mean estimation, which are of a similar form. Up to constant factors, these lower bounds show that the simple strategy of either treating all data as private or discarding the private data, is optimal. We also study PA-DP supervised learning with \textit{unlabeled} public samples. In contrast to our previous result, we here show novel methods for leveraging public data in private supervised learning. For generalized linear models (GLM) with unlabeled public data, we show an efficient algorithm which, given $\tilde{O}({n_{\text{priv}}}\epsilon)$ unlabeled public samples, achieves the dimension independent rate $\tilde{O}\big(\frac{1}{\sqrt{{n_{\text{priv}}}}} + \frac{1}{\sqrt{{n_{\text{priv}}}\epsilon}}\big)$. We develop new lower bounds for this setting which shows that this rate cannot be improved with more public samples, and any fewer public samples leads to a worse rate. Finally, we provide extensions of this result to general hypothesis classes with finite \textit{fat-shattering dimension} with applications to neural networks and non-Euclidean geometries.

We focus on developing a theoretical understanding of meta-learning. Given multiple tasks drawn i.i.d. from some (unknown) task distribution, the goal is to find a good pre-trained model that can be adapted to a new, previously unseen, task with little computational and statistical overhead. We introduce a novel notion of stability for meta-learning algorithms, namely *uniform meta-stability*. We instantiate two uniformly meta-stable learning algorithms based on regularized empirical risk minimization and gradient descent and give explicit generalization bounds for convex learning problems with smooth losses and for weakly convex learning problems with non-smooth losses. Finally, we extend our results to stochastic and adversarially robust variants of our meta-learning algorithm.

We study adversarially robust transfer learning, wherein, given labeled data on multiple (source) tasks, the goal is to train a model with small robust error on a previously unseen (target) task. In particular, we consider a multi-task representation learning (MTRL) setting, i.e., we assume that the source and target tasks admit a simple (linear) predictor on top of a shared representation (e.g., the final hidden layer of a deep neural network). In this general setting, we provide rates on~the excess adversarial (transfer) risk for Lipschitz losses and smooth nonnegative losses. These rates show that learning a representation using adversarial training on diverse tasks helps protect against inference-time attacks in data-scarce environments. Additionally, we provide novel rates for the single-task setting.

Adversarial training has emerged as a popular approach for training models that are robust to inference-time adversarial attacks. However, our theoretical understanding of why and when it works remains limited. Prior work has offered generalization analysis of adversarial training, but they are either restricted to the Neural Tangent Kernel (NTK) regime or they make restrictive assumptions about data such as (noisy) linear separability or robust realizability. In this work, we study the stability and generalization of adversarial training for two-layer networks **without any data distribution assumptions** and **beyond the NTK regime**. Our findings suggest that for networks with *any given initialization* and *sufficiently large width*, the generalization bound can be effectively controlled via early stopping. We further improve the generalization bound by leveraging smoothing using Moreau’s envelope.

Standard deep reinforcement learning (DRL) aims to maximize expected reward, considering collected experiences equally in formulating a policy. This differs from human decision-making, where gains and losses are valued differently and outlying outcomes are given increased consideration. It also fails to capitalize on opportunities to improve safety and/or performance through the incorporation of distributional context. Several approaches to distributional DRL have been investigated, with one popular strategy being to evaluate the projected distribution of returns for possible actions. We propose a more direct approach whereby risk-sensitive objectives, specified in terms of the cumulative distribution function (CDF) of the distribution of full-episode rewards, are optimized. This approach allows for outcomes to be weighed based on relative quality, can be used for both continuous and discrete action spaces, and may naturally be applied in both constrained and unconstrained settings. We show how to compute an asymptotically consistent estimate of the policy gradient for a broad class of risk-sensitive objectives via sampling, subsequently incorporating variance reduction and regularization measures to facilitate effective on-policy learning. We then demonstrate that the use of moderately "pessimistic" risk profiles, which emphasize scenarios where the agent performs poorly, leads to enhanced exploration and a continual focus on addressing deficiencies. We test the approach using different risk profiles in six OpenAI Safety Gym environments, comparing to state of the art on-policy methods. Without cost constraints, we find that pessimistic risk profiles can be used to reduce cost while improving total reward accumulation. With cost constraints, they are seen to provide higher positive rewards than risk-neutral approaches at the prescribed allowable cost.

Sample-efficient offline reinforcement learning (RL) with linear function approximation has been studied extensively recently. Much of the prior work has yielded instance-independent rates that hold even for the worst-case realization of problem instances. This work seeks to understand instance-dependent bounds for offline RL with linear function approximation. We present an algorithm called Bootstrapped and Constrained Pessimistic Value Iteration (BCP-VI), which leverages data bootstrapping and constrained optimization on top of pessimism. We show that under a partial data coverage assumption, that of concentrability with respect to an optimal policy, the proposed algorithm yields a fast rate for offline RL when there is a positive gap in the optimal Q-value functions, even if the offline data were collected adaptively. Moreover, when the linear features of the optimal actions in the states reachable by an optimal policy span those reachable by the behavior policy and the optimal actions are unique, offline RL achieves absolute zero sub-optimality error when the number of episodes exceeds a (finite) instance-dependent threshold. To the best of our knowledge, these are the first results that give a fast rate bound on the sub-optimality and an absolute zero sub-optimality bound for offline RL with linear function approximation from adaptive data with partial coverage. We also provide instance-agnostic and instance-dependent information-theoretical lower bounds to complement our upper bounds.

We study the problem of approximating stationary points of Lipschitz and smooth functions under $(\varepsilon,\delta)$-differential privacy (DP) in both the finite-sum and stochastic settings. A point $\widehat{w}$ is called an $\alpha$-stationary point of a function $F:\mathbb{R}^d\rightarrow\mathbb{R}$ if $\|\nabla F(\widehat{w})\|\leq \alpha$. We give a new construction that improves over the existing rates in the stochastic optimization setting, where the goal is to find approximate stationary points of the population risk given $n$ samples. Our construction finds a $\tilde{O}\big(\frac{1}{n^{1/3}} + \big[\frac{\sqrt{d}}{n\varepsilon}\big]^{1/2}\big)$-stationary point of the population risk in time linear in $n$. We also provide an efficient algorithm that finds an $\tilde{O}\big(\big[\frac{\sqrt{d}}{n\varepsilon}\big]^{2/3}\big)$-stationary point in the finite-sum setting. This improves on the previous best rate of $\tilde{O}\big(\big[\frac{\sqrt{d}}{n\varepsilon}\big]^{1/2}\big)$. Furthermore, under the additional assumption of convexity, we completely characterize the sample complexity of finding stationary points of the population risk (up to polylog factors) and show that the optimal rate on population stationarity is $\tilde \Theta\big(\frac{1}{\sqrt{n}}+\frac{\sqrt{d}}{n\varepsilon}\big)$. Finally, we show that our methods can be used to provide dimension-independent rates of $O\big(\frac{1}{\sqrt{n}}+\min\big(\big[\frac{\sqrt{rank}}{n\varepsilon}\big]^{2/3},\frac{1}{(n\varepsilon)^{2/5}}\big)\big)$ on population stationarity for Generalized Linear Models (GLM), where $rank$ is the rank of the design matrix, which improves upon the previous best known rate.

We formalize the problem of machine unlearning as design of efficient unlearning algorithms corresponding to learning algorithms which perform a selection of adaptive queries from structured query classes. We give efficient unlearning algorithms for linear and prefix-sum query classes. As applications, we show that unlearning in many problems, in particular, stochastic convex optimization (SCO), can be reduced to the above, yielding improved guarantees for the problem. In particular, for smooth Lipschitz losses and any $\rho>0$, our results yield an unlearning algorithm with excess population risk of $\tilde O\big(\frac{1}{\sqrt{n}}+\frac{\sqrt{d}}{n\rho}\big)$ with unlearning query (gradient) complexity $\tilde O(\rho \cdot \text{Retraining Complexity})$, where $d$ is the model dimensionality and $n$ is the initial number of samples. For non-smooth Lipschitz losses, we give an unlearning algorithm with excess population risk $\tilde O\big(\frac{1}{\sqrt{n}}+\big(\frac{\sqrt{d}}{n\rho}\big)^{1/2}\big)$ with the same unlearning query (gradient) complexity. Furthermore, in the special case of Generalized Linear Models (GLMs), such as those in linear and logistic regression, we get dimension-independent rates of $\tilde O\big(\frac{1}{\sqrt{n}} +\frac{1}{(n\rho)^{2/3}}\big)$ and $\tilde O\big(\frac{1}{\sqrt{n}} +\frac{1}{(n\rho)^{1/3}}\big)$ for smooth Lipschitz and non-smooth Lipschitz losses respectively. Finally, we give generalizations of the above from one unlearning request to dynamic streams consisting of insertions and deletions.

As trained intelligent systems become increasingly pervasive, multiagent learning has emerged as a popular framework for studying complex interactions between autonomous agents. Yet, a formal understanding of how and when learners in heterogeneous environments benefit from sharing their respective experiences is far from complete. In this paper, we seek answers to these questions in the context of linear contextual bandits. We present a novel distributed learning algorithm based on the upper confidence bound (UCB) algorithm, which we refer to as H-LINUCB, wherein agents cooperatively minimize the group regret under the coordination of a central server. In the setting where the level of heterogeneity or dissimilarity across the environments is known to the agents, we show that H-LINUCB is provably optimal in regimes where the tasks are highly similar or highly dissimilar.

We study robust adversarial training of two-layer neural networks as a bi-level optimization problem. In particular, for the inner loop that implements the adversarial attack during training using projected gradient descent (PGD), we propose maximizing a \emph{lower bound} on the $0/1$-loss by reflecting a surrogate loss about the origin. This allows us to give a convergence guarantee for the inner-loop PGD attack. Furthermore, assuming the data is linearly separable, we provide precise iteration complexity results for end-to-end adversarial training, which holds for any width and initialization. We provide empirical evidence to support our theoretical results.

We seek to understand what facilitates sample-efficient learning from historical datasets for sequential decision-making, a problem that is popularly known as offline reinforcement learning (RL). Further, we are interested in algorithms that enjoy sample efficiency while leveraging (value) function approximation. In this paper, we address these fundamental questions by (i) proposing a notion of data diversity that subsumes the previous notions of coverage measures in offline RL and (ii) using this notion to \emph{unify} three distinct classes of offline RL algorithms based on version spaces (VS), regularized optimization (RO), and posterior sampling (PS). We establish that VS-based, RO-based, and PS-based algorithms, under standard assumptions, achieve \emph{comparable} sample efficiency, which recovers the state-of-the-art sub-optimality bounds for finite and linear model classes with the standard assumptions. This result is surprising, given that the prior work suggested an unfavorable sample complexity of the RO-based algorithm compared to the VS-based algorithm, whereas posterior sampling is rarely considered in offline RL due to its explorative nature. Notably, our proposed model-free PS-based algorithm for offline RL is \emph{novel}, with sub-optimality bounds that are \emph{frequentist} (i.e., worst-case) in nature.

We study the problem of transfer learning via Multi-Task Representation Learning (MTRL), wherein multiple source tasks are used to learn a good common representation, and a predictor is trained on top of it for the target task. Under standard regularity assumptions on the loss function and task diversity, we provide new statistical rates on the excess risk of the target task, which demonstrate the benefit of representation learning. Importantly, our rates are optimistic, i.e., they interpolate between the standard $O(m^{-1/2})$ rate and the fast $O(m^{-1})$ rate, depending on the difficulty of the learning task, where $m$ is the number of samples for the target task. Besides the main result, we make several new contributions, including giving optimistic rates for excess risk of source tasks (multi-task learning (MTL)), a local Rademacher complexity theorem for MTRL and MTL, as well as a chain rule for local Rademacher complexity for composite predictor classes.

We study the problem of $(\epsilon,\delta)$-differentially private learning of linear predictors with convex losses. We provide results for two subclasses of loss functions. The first case is when the loss is smooth and non-negative but not necessarily Lipschitz (such as the squared loss). For this case, we establish an upper bound on the excess population risk of $\tilde{O}\left(\frac{\Vert w^*\Vert}{\sqrt{n}} + \min\left\{\frac{\Vert w^* \Vert^2}{(n\epsilon)^{2/3}},\frac{\sqrt{d}\Vert w^*\Vert^2}{n\epsilon}\right\}\right)$, where $n$ is the number of samples, $d$ is the dimension of the problem, and $w^*$ is the minimizer of the population risk. Apart from the dependence on $\Vert w^\ast\Vert$, our bound is essentially tight in all parameters. In particular, we show a lower bound of $\tilde{\Omega}\left(\frac{1}{\sqrt{n}} + {\min\left\{\frac{\Vert w^*\Vert^{4/3}}{(n\epsilon)^{2/3}}, \frac{\sqrt{d}\Vert w^*\Vert}{n\epsilon}\right\}}\right)$. We also revisit the previously studied case of Lipschitz losses \cite{SSTT21}. For this case, we close the gap in the existing work and show that the optimal rate is (up to log factors) $\Theta\left(\frac{\Vert w^*\Vert}{\sqrt{n}} + \min\left\{\frac{\Vert w^*\Vert}{\sqrt{n\epsilon}},\frac{\sqrt{\text{rank}}\Vert w^*\Vert}{n\epsilon}\right\}\right)$, where $\text{rank}$ is the rank of the design matrix. This improves over existing work in the high privacy regime. Finally, our algorithms involve a private model selection approach that we develop to enable attaining the stated rates without a-priori knowledge of $\Vert w^*\Vert$.

Recent works show that adversarial examples exist for random neural networks [Daniely and Schacham, 2020] and that these examples can be found using a single step of gradient ascent [Bubeck et al., 2021]. In this work, we extend this line of work to ``lazy training'' of neural networks -- a dominant model in deep learning theory in which neural networks are provably efficiently learnable. We show that over-parametrized neural networks that are guaranteed to generalize well and enjoy strong computational guarantees remain vulnerable to attacks generated using a single step of gradient ascent.

We study the problem of corralling stochastic bandit algorithms, that is combining multiple bandit algorithms designed for a stochastic environment, with the goal of devising a corralling algorithm that performs almost as well as the best base algorithm. We give two general algorithms for this setting, which we show benefit from favorable regret guarantees. We show that the regret of the corralling algorithms is no worse than that of the best algorithm containing the arm with the highest reward, and depends on the gap between the highest reward and other rewards.

In this paper, we focus on answering queries, in a differentially private manner, on graph streams. We adopt the sliding window model of privacy, where we wish to perform analysis on the last $W$ updates and ensure that privacy is preserved for the entire stream. We show that in this model, the price of ensuring differential privacy is minimal. Furthermore, since differential privacy is preserved under post-processing, our results can be used as a subroutine in many tasks, including Lipschitz learning on graphs, cut functions, and spectral clustering.

We study the problem of machine unlearning and identify a notion of algorithmic stability, Total Variation (TV) stability, which we argue, is suitable for the goal of exact unlearning. For convex risk minimization problems, we design TV-stable algorithms based on noisy Stochastic Gradient Descent (SGD). Our key contribution is the design of corresponding efficient unlearning algorithms, which are based on constructing a near-maximal coupling of Markov chains for the noisy SGD procedure. To understand the trade-offs between accuracy and unlearning efficiency, we give upper and lower bounds on excess empirical and populations risk of TV stable algorithms for convex risk minimization. Our techniques generalize to arbitrary non-convex functions, and our algorithms are differentially private as well.

We investigate the capacity control provided by dropout in various machine learning problems. First, we study dropout for matrix completion, where it induces a distribution-dependent regularizer that equals the weighted trace-norm of the product of the factors. In deep learning, we show that the distribution-dependent regularizer due to dropout directly controls the Rademacher complexity of the underlying class of deep neural networks. These developments enable us to give concrete generalization error bounds for the dropout algorithm in both matrix completion as well as training deep neural networks.

We investigate the robustness of stochastic approximation approaches against data poisoning attacks. We focus on two-layer neural networks with ReLU activation and show that under a specific notion of separability in the RKHS induced by the infinite-width network, training (finite-width) networks with stochastic gradient descent is robust against data poisoning attacks. Interestingly, we find that in addition to a lower bound on the width of the network, which is standard in the literature, we also require a distribution-dependent upper bound on the width for robust generalization. We provide extensive empirical evaluations that support and validate our theoretical results.

Existing approaches to federated learning suffer from a communication bottleneck as well as convergence issues due to sparse client participation. In this paper we introduce a novel algorithm,called FetchSGD, to overcome these challenges. FetchSGD compresses model updates using a Count Sketch, and then takes advantage of the mergeability of sketches to combine model updates from many workers. A key insight in the design of FetchSGD is that, because the Count Sketch is linear, momentum and error accumulation can both be carried out within the sketch.This allows the algorithm to move momentum and error accumulation from clients to the central aggregator, overcoming the challenges of sparse client participation while still achieving high compression rates and good convergence. We prove that FetchSGD has favorable convergence guarantees, and we demonstrate its empirical effectiveness by training two residual networks and a transformer model.

We study dropout in two-layer neural networks with rectified linear unit (ReLU) activations. Under mild overparametrization and assuming that the limiting kernel can separate the data distribution with a positive margin, we show that the dropout training with logistic loss achieves $\epsilon$-suboptimality in the test error in $O(1/\epsilon)$ iterations.

Several recent results provide theoretical insights into the phenomena of adversarial examples. Existing results, however, are often limited due to a gap between the simplicity of the models studied and the complexity of those deployed in practice. In this work, we strike a better balance by considering a model that involves learning a representation while at the same time giving a precise generalization bound and a robustness certificate. We focus on the hypothesis class obtained by combining a sparsity-promoting encoder coupled with a linear classifier, and show an interesting interplay between the expressivity and stability of the (supervised) representation map and a notion of margin in the feature space. We bound the robust risk (to $\ell_2$-bounded perturbations) of hypotheses parameterized by dictionaries that achieve a mild encoder gap on training data. Furthermore, we provide a robustness certificate for end-to-end classification. We demonstrate the applicability of our analysis by computing certified accuracy on real data, and compare with other alternatives for certified robustness.

An increasing number of datasets contain multiple views, such as video, sound and automatic captions. A basic challenge in representation learning is how to leverage multiple views to learn better representations. This is further complicated by the existence of a latent alignment between views, such as between speech and its transcription, and by the multitude of choices for the learning objective. We explore an advanced, correlation-based representation learning method on a 4-way parallel, multimodal dataset, and assess the quality of the learned representations on retrieval-based tasks. We show that the proposed approach produces rich representations that capture most of the information shared across views. Our best models for speech and textual modalities achieve retrieval rates from 70.7% to 96.9% on open-domain, user-generated instructional videos. This shows it is possible to learn reliable representations across disparate, unaligned and noisy modalities, and encourages using the proposed approach on larger datasets.

We give a formal and complete characterization of the explicit regularizer induced by dropout in deep linear networks with squared loss. We show that (a) the explicit regularizer is composed of an $\ell_2$-path regularizer and other terms that are also re-scaling invariant, (b) the convex envelope of the induced regularizer is the squared nuclear norm of the network map, and (c) for a sufficiently large dropout rate, we characterize the global optima of the dropout objective. We validate our theoretical findings with empirical results.

Prospection is key to solving challenging problems in new environments, but it has not been deeply explored as applied to task planning for perception-driven robotics. We propose visual robot task planning, where we take in an input image and must generate a sequence of high-level actions and associated observations that achieve some task. In this paper, we describe a neural network architecture and associated planning algorithm that (1) learns a representation of the world that can generate prospective futures, (2) uses this generative model to simulate the result of sequences of high-level actions in a variety of environments, and (3) evaluates these actions via a variant of Monte Carlo Tree Search to find a viable solution to a particular problem. Our approach allows us to visualize intermediate motion goals and learn to plan complex activity from visual information, and used this to generate and visualize task plans on held-out examples of a block-stacking simulation.

We revisit two algorithms, matrix stochastic gradient (MSG) and $\ell_2$-regularized MSG (RMSG), that are instances of stochastic gradient descent (SGD) on a convex relaxation to principal component analysis (PCA). These algorithms have been shown to outperform Oja’s algorithm, empirically, in terms of the iteration complexity, and to have runtime comparable with Oja’s. However, these findings are not supported by existing theoretical results. While the iteration complexity bound for $\ell_2$-RMSG was recently shown to match that of Oja’s algorithm, its theoretical efficiency was left as an open problem. In this work, we give improved bounds on per iteration cost of mini-batched variants of both MSG and $\ell_2$-RMSG and arrive at an algorithm with total computational complexity matching that of Oja's algorithm.

We study the adversarial multi-armed bandit problem where the learner is supplied with partial observations modeled by a \emph{feedback graph} and where shifting to a new action incurs a fixed \emph{switching cost}. We give two new algorithms for this problem in the informed setting. Our best algorithm achieves a pseudo-regret of $\tilde O(\gamma(G)^{\frac{1}{3}}T^{\frac{2}{3}})$, where $\gamma(G)$ is the domination number of the feedback graph. This significantly improves upon the previous best result for the same problem, which was based on the independence number of $G$. We also present matching lower bounds for our result that we describe in detail. Finally, we give a new algorithm with improved policy regret bounds when partial counterfactual feedback is available.

In this paper, we study private sparsification of graphs. In particular, we give an algorithm that given an input graph, returns a sparse graph which approximates the spectrum of the input graph while ensuring differential privacy. This allows one to solve many graph problems privately yet efficiently and accurately. This is exemplified with application of the proposed meta-algorithm to graph algorithms for privately answering cut-queries, as well as practical algorithms for computing {\scshape MAX-CUT} and {\scshape SPARSEST-CUT} with better accuracy than previously known. We also give the first efficient private algorithm to learn Laplacian eigenmap on a graph.

Large-scale distributed training of neural networks is often limited by network bandwidth, wherein the communication time overwhelms the local computation time. Motivated by the success of sketching methods in sub-linear/streaming algorithms, we introduce Sketched-SGD, an algorithm for carrying out distributed SGD by communicating sketches instead of full gradients. We show that \ssgd has favorable convergence rates on several classes of functions. When considering all communication -- both of gradients and of updated model weights -- Sketched-SGD reduces the amount of communication required compared to other gradient compression methods from $\mathcal{O}(d)$ or $\mathcal{O}(W)$ to $\mathcal{O}(\log d)$, where $d$ is the number of model parameters and $W$ is the number of workers participating in training. We run experiments on a transformer model, an LSTM, and a residual network, demonstrating up to a 40x reduction in total communication cost with no loss in final model performance. We also show experimentally that Sketched-SGD scales to at least 256 workers without increasing communication cost or degrading model performance.

We propose a general theory for studying the landscape of nonconvex optimization with underlying symmetric structures for a class of machine learning problems (e.g., low-rank matrix factorization, phase retrieval, and deep linear neural networks). In particular, we characterize the locations of stationary points and the null space of Hessian matrices of the objective function via the lens of invariant groups. As a major motivating example, we apply the proposed general theory to characterize the global landscape of the nonconvex optimization in low-rank matrix factorization problem. We illustrate how the rotational symmetry group gives rise to infinitely many nonisolated strict saddle points and equivalent global minima of the objective function. By explicitly identifying all stationary points, we divide the entire parameter space into three regions: (R1) the region containing the neighborhoods of all strict saddle points where the objective has negative curvature; (R2) the region containing neighborhoods of all global minima, where the objective enjoys strong convexity along certain directions; and (R3) the complement of the above regions, where the gradient has sufficiently large magnitude. We further extend our result to the matrix sensing problem. Such global landscape implies that strong global convergence guarantees for popular iterative algorithms with arbitrary initial solutions.

Algorithmic approaches endow deep learning systems with implicit bias that helps them generalize even in over-parametrized settings. In this paper, we focus on understanding such a bias induced in learning through dropout, a popular technique to avoid overfitting in deep learning. For single hidden-layer linear neural networks, we show that dropout tends to make the norm of incoming/outgoing weight vectors of all the hidden nodes equal. In addition, we provide a complete characterization of the optimization landscape induced by dropout.

In this paper, we study the following robust low-rank matrix approximation problem: given a matrix $A \in \R^{n \times d}$, find a rank-$k$ matrix $B$, while satisfying differential privacy, such that $ \norm{ A - B }_p \leq \alpha \mathsf{OPT}_k(A) + \tau,$ where $\norm{ M }_p$ is the entry-wise $\ell_p$-norm and $\mathsf{OPT}_k(A):=\min_{\mathsf{rank}(X) \leq k} \norm{ A - X}_p$. It is well known that low-rank approximation w.r.t. entrywise $\ell_p$-norm, for $p \in [1,2)$, yields robustness to gross outliers in the data. We propose an algorithm that guarantees $\alpha=\widetilde{O}(k^2), \tau=\widetilde{O}(k^2(n+kd)/\varepsilon)$, runs in $\widetilde O((n+d)\poly~k)$ time and uses $O(k(n+d)\log k)$ space. We study extensions to the streaming setting where entries of the matrix arrive in an arbitrary order and output is produced at the very end or continually. We also study the related problem of differentially private robust principal component analysis (PCA), wherein we return a rank-$k$ projection matrix $\Pi$ such that $\norm{ A - A \Pi }_p \leq \alpha \mathsf{OPT}_k(A) + \tau.$

The notion of policy regret'' in online learning is supposed to capture the reactions of the adversary to the actions taken by the learner, which more traditional notions such as external regret do not take into account. We revisit this notion of policy regret, and first show that there are online learning settings in which policy regret and external regret are incompatible: any sequence of play which does well with respect to one must do poorly with respect to the other. We then focus on the game theoretic setting, when the adversary is a self-interested agent. In this setting we show that the external regret and policy regret are not in conflict, and in fact that a wide class of algorithms can ensure both as long as the adversary is also using such an algorithm. We also define a new notion of equilibrium which we call apolicy equilibrium'', and show that no-policy regret algorithms will have play which converges to such an equilibrium. Relating this back to external regret, we show that coarse correlated equilibria (which no-external regret players will converge to) are a strict subset of policy equilibria. So in game-theoretic settings every sequence of play with no external regret also has no policy regret, but the converse is not true.

We study the statistical and computational aspects of kernel principal component analysis using random Fourier features and show that under mild assumptions, $O(\sqrt{n} \log n)$ features suffices to achieve $O(1/\epsilon^2)$ sample complexity. Furthermore, we give a memory efficient streaming algorithm based on classical Oja's algorithm that achieves this rate

We use differential equations based approaches to provide some {\it \textbf{physics}} insights into analyzing the dynamics of popular optimization algorithms in machine learning. In particular, we study gradient descent, proximal gradient descent, coordinate gradient descent, proximal coordinate gradient, and Newton's methods as well as their Nesterov's accelerated variants in a unified framework motivated by a natural connection of optimization algorithms to physical systems. Our analysis is applicable to more general algorithms and optimization problems {\it \textbf{beyond}} convexity and strong convexity, e.g. Polyak-\L ojasiewicz and error bound conditions (possibly nonconvex).

Different modes of vibration of the vocal folds contribute significantly to the voice quality. The neutral mode phonation, often used in a modal voice, is one against which the other modes can be contrastively described, also called non-modal phonations. This paper investigates the impact of non-modal phonation on phonological posteriors, the probabilities of phonological features inferred from the speech signal using a deep learning approach. Five different non-modal phonations are considered: falsetto, creaky, harshness, tense and breathiness. The impact of such non-modal phonation on phonological features, the Sound Patterns of English (SPE), is investigated in both speech analysis and synthesis tasks. We found that breathy and tense phonation impact the SPE features less, creaky phonation impacts the features moderately, and harsh and falsetto phonation impact the phonological features the most. We also report invariant and the most different SPE features impacted by non-modal phonation.

Information from different bio-signals such as speech, handwriting, and gait have been used to monitor the state of Parkinson's disease (PD) patients, however, all the multimodal bio-signals may not always be available. We propose a method based on multi-view representation learning via generalized canonical correlation analysis (GCCA) for learning a representation of features extracted from handwriting and gait that can be used as a complement to speech-based features. Three different problems are addressed: classification of PD patients vs. healthy controls, prediction of the neurological state of PD patients according to the UPDRS score, and the prediction of a modified version of the Frenchay dysarthria assessment (m-FDA). According to the results, the proposed approach is suitable to improve the results in the addressed problems, specially in the prediction of the UPDRS, and m-FDA scores.

International audience

We propose novel first-order stochastic approximation algorithms for canonical correlation analysis (CCA). Algorithms presented are instances of inexact matrix stochastic gradient (MSG) and inexact matrix exponentiated gradient (MEG), and achieve $\epsilon$-suboptimality in the population objective in $\operatorname{poly}(\frac{1}{\epsilon})$ iterations. We also consider practical variants of the proposed algorithms and compare them with other methods for CCA both theoretically and empirically.

The cyclic block coordinate descent-type (CBCD-type) methods have shown remarkable computational performance for solving strongly convex minimization problems. Typical applications include many popular statistical machine learning methods such as elastic-net regression, ridge penalized logistic regression, and sparse additive regression. Existing optimization literature has shown that the CBCD-type methods attain iteration complexity of O(p⋅\log(1/ε)), where εis a pre-specified accuracy of the objective value, and p is the number of blocks. However, such iteration complexity explicitly depends on p, and therefore is at least p times worse than those of gradient descent methods. To bridge this theoretical gap, we propose an improved convergence analysis for the CBCD-type methods. In particular, we first show that for a family of quadratic minimization problems, the iteration complexity of the CBCD-type methods matches that of the GD methods in term of dependency on p (up to a \log^2 p factor). Thus our complexity bounds are sharper than the existing bounds by at least a factor of p/\log^2p. We also provide a lower bound to confirm that our improved complexity bounds are tight (up to a \log^2 p factor) if the largest and smallest eigenvalues of the Hessian matrix do not scale with p. Finally, we generalize our analysis to other strongly convex minimization problems beyond quadratic ones.

Partial Least Squares (PLS) is a ubiquitous statistical technique for bilinear factor analysis. It is used in many data analysis, machine learning, and information retrieval applications to model the covariance structure between a pair of data matrices. In this paper, we consider PLS for representation learning in a multiview setting where we have more than one view in data at training time. Furthermore, instead of framing PLS as a problem about a fixed given data set, we argue that PLS should be studied as a stochastic optimization problem, especially in a "big data" setting, with the goal of optimizing a population objective based on sample. This view suggests using Stochastic Approximation (SA) approaches, such as Stochastic Gradient Descent (SGD) and enables a rigorous analysis of their benefits. In this paper, we develop SA approaches to PLS and provide iteration complexity bounds for the proposed algorithms.

We propose a stochastic variance reduced optimization algorithm for solving a class of large-scale nonconvex optimization problems with cardinality constraints, and provide sufficient conditions under which the proposed algorithm enjoys strong linear convergence guarantees and optimal estimation accuracy in high dimensions. Numerical experiments demonstrate the efficiency of our method in terms of both parameter estimation and computational performance.

Mo Yu, Mark Dredze, Raman Arora, Matthew R. Gormley. Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. 2016.

Medical researchers are coming to appreciate that many diseases are in fact complex, heterogeneous syndromes composed of subpopulations that express different variants of a related complication. Longitudinal data extracted from individual electronic health records (EHR) offer an exciting new way to study subtle differences in the way these diseases progress over time. In this paper, we focus on answering two questions that can be asked using these databases of longitudinal EHR data. First, we want to understand whether there are individuals with similar disease trajectories and whether there are a small number of degrees of freedom that account for differences in trajectories across the population. Second, we want to understand how important clinical outcomes are associated with disease trajectories. To answer these questions, we propose the Disease Trajectory Map (DTM), a novel probabilistic model that learns low-dimensional representations of sparse and irregularly sampled longitudinal data. We propose a stochastic variational inference algorithm for learning the DTM that allows the model to scale to large modern medical datasets. To demonstrate the DTM, we analyze data collected on patients with the complex autoimmune disease, scleroderma. We find that DTM learns meaningful representations of disease trajectories and that the representations are significantly associated with important clinical outcomes.

It has been previously shown that, when both acoustic and articulatory training data are available, it is possible to improve phonetic recognition accuracy by learning acoustic features from this multi-view data with canonical correlation analysis (CCA). In contrast with previous work based on linear or kernel CCA, we use the recently proposed deep CCA, where the functional form of the feature mapping is a deep neural network. We apply the approach on a speaker-independent phonetic recognition task using data from the University of Wisconsin X-ray Microbeam Database. Using a tandem-style recognizer on this task, deep CCA features improve over earlier multi-view approaches as well as over articulatory inversion and typical neural network-based tandem features. We also present a new stochastic training approach for deep CCA, which produces both faster training and better-performing features.

We consider learning representations (features) in the setting in which we have access to multiple unlabeled views of the data for representation learning while only one view is available at test time. Previous work on this problem has proposed several techniques based on deep neural networks, typically involving either autoencoder-like networks with a reconstruction objective or paired feedforward networks with a correlation-based objective. We analyze several techniques based on prior work, as well as new variants, and compare them experimentally on visual, speech, and language domains. To our knowledge this is the first head-to-head comparison of a variety of such techniques on multiple tasks. We find an advantage for correlation-based representation learning, while the best results on most tasks are obtained with our new variant, deep canonically correlated autoencoders (DCCAE).

Pushpendre Rastogi, Benjamin Van Durme, Raman Arora. Proceedings of the 2015 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. 2015.

We consider the problem of finding lower dimensional subspaces in the presence of outliers and noise in the online setting. In particular, we extend previous batch formulations of robust PCA to the stochastic setting with minimal storage requirements and runtime complexity. We introduce three novel stochastic approximation algorithms for robust PCA that are extensions of standard algorithms for PCA - the stochastic power method, incremental PCA and online PCA using matrix-exponentiated-gradient (MEG) updates. For robust online PCA we also give a sub-linear convergence guarantee. Our numerical results demonstrate the superiority of the the robust online method over the other robust stochastic methods and the advantage of robust methods over their non-robust counterparts in the presence of outliers in artificial and real scenarios.

Previous work has shown that acoustic features can be improved by unsupervised learning of transformations based on canonical correlation analysis (CCA) using articulatory measurements that are available at training time. In this paper, we investigate whether this second view (articulatory data) still helps even when labels are also available at training time. We begin with strong baseline bottleneck features, which can be learned when the training set is phonetically labeled. We then compare several options for learning transformations of the bottleneck features in the presence of both articulatory measurements and phonetic labels for the training data. The methods compared include combinations of LDA and CCA, as well as a three-view extension of CCA that simultaneously uses the labels and articulatory measurements as additional views. Phonetic recognition experiments on data from the University of Wisconsin X-ray microbeam database show that the learned features improve performance over using either just the labels or just the articulatory measurements for learning acoustic transformations.

Articulatory measurements have been used in a variety of speech science and technology applications. These measurements can be obtained with a number of technologies, such as electromagnetic articulography and X-ray microbeam, typically involving pellets attached to individual articulators. Due to limitations in the recording technologies, articulatory measurements often contain missing data when individual pellets are mis-tracked, leading to relatively high rates of loss in this expensive and time-consuming data source. We present an approach to reconstructing such data, using low-rank matrix factorization techniques combined with temporal smoothness regularization, and apply it to reconstructing the missing entries in the Wisconsin X-ray microbeam database. Our algorithm alternates between two simple steps, each having a closed form as the solution of a linear system. The algorithm gives realistic reconstructions even when a majority of the frames contain missing data, improving over previous approaches to this problem in terms of both root mean squared error and phonetic recognition performance when using the reconstructions.

Signed permutations (also known as the hyperoctahedral group) are used in modeling genome rearrangements. The algorithmic problems they raise are computationally demanding when not NP-hard. This paper presents a tractable algorithm for learning consensus ranking between signed permutations under the inversion distance. This can be extended to estimate a natural class of exponential models over the group of signed permutations. We investigate experimentally the efficiency of our algorithm for modeling data generated by random reversals.

Canonical correlation analysis (CCA) and kernel CCA can be used for unsupervised learning of acoustic features when a second view (e.g., articulatory measurements) is available for some training data, and such projections have been used to improve phonetic frame classification. Here we study the behavior of CCA-based acoustic features on the task of phonetic recognition, and investigate to what extent they are speaker-independent or domain-independent. The acoustic features are learned using data drawn from the University of Wisconsin X-ray Microbeam Database (XRMB). The features are evaluated within and across speakers on XRMB data, as well as on out-of-domain TIMIT and MOCHA-TIMIT data. Experimental results show consistent improvement with the learned acoustic features over baseline MFCCs and PCA projections. In both speaker-dependent and cross-speaker experiments, phonetic error rates are improved by 4-9% absolute (10-23% relative) using CCA-based features over baseline MFCCs. In cross-domain phonetic recognition (training on XRMB and testing on MOCHA or TIMIT), the learned projections provide smaller improvements.

We introduce Deep Canonical Correlation Analysis (DCCA), a method to learn complex nonlinear transformations of two views of data such that the resulting representations are highly linearly correlated. Parameters of both transformations are jointly learned to maximize the (regularized) total correlation. It can be viewed as a nonlinear extension of the linear method \emphcanonical correlation analysis (CCA). It is an alternative to the nonparametric method \emphkernel canonical correlation analysis (KCCA) for learning correlated nonlinear transformations. Unlike KCCA, DCCA does not require an inner product, and has the advantages of a parametric method: training time scales well with data size and the training data need not be referenced when computing the representations of unseen instances. In experiments on two real-world datasets, we find that DCCA learns representations with significantly higher correlation than those learned by CCA and KCCA. We also introduce a novel non-saturating sigmoid function based on the cube root that may be useful more generally in feedforward neural networks.

We study PCA as a stochastic optimization problem and propose a novel stochastic approximation algorithm which we refer to as Matrix Stochastic Gradient'' (MSG), as well as a practical variant, Capped MSG. We study the method both theoretically and empirically. "

In many applications of regression, one is concerned with the efficiency of the estimated function in addition to the accuracy of the regression. For efficiency, it is common to represent the estimated function as a rectangular lattice of values-a lookup table (LUT)-that can be linearly interpolated for any needed value. Typically, a LUT is constructed from data with a two-step process that first fits a function to the data, then evaluates that fitted function at the nodes of the lattice. We present an approach, termed lattice regression, that directly optimizes the values of the lattice nodes to minimize the post-interpolation training error. Additionally, we propose a second-order difference regularizer to promote smoothness. We demonstrate the effectiveness of this approach on two image processing tasks that require both accurate regression and efficient function evaluations: inverse device characterization for color management and omnidirectional super-resolution for visual homing.

This paper analyses the computational complexity and stability of an online algorithm recently proposed for learning rotations. The proposed algorithm involves multiplicative updates that are matrix exponentials of skew-symmetric matrices comprising the Lie algebra of the rotation group. The rank-deficiency of the skew-symmetric matrices involved in the updates is exploited to reduce the updates to a simple quadratic form. The Lyapunov stability of the algorithm is established and the application of the algorithm to registration of point-clouds in n-dimensional Euclidean space is discussed.

The joint invariants of the projective group PSL(3,R) on R2, the five-point volume cross-ratios, are studied to address the problem of correspondence in a camera network. The distribution of cross-ratios over the unit square as well as in a small local-neighbourhood of a reference point are found to have a heavy tail. No cross ratio value is unique but the collection of five point cross ratios generated by taking all possible combination of five points completely prescribes the curve. Sections of the signature submanifold that admit large enough variation of cross ratios are found to be sufficient in providing correspondence across wide perspectives. Such invariant signatures may be collected independently at cameras with different viewpoints and shared, thereby achieving the registration of objects in the image. Experimental results with license plate database are provided.

An algorithm is presented for online learning of rotations. The proposed algorithm involves matrix exponentiated gradient updates and is motivated by the Von Neumann divergence. The additive updates are skew-symmetric matrices with trace zero which comprise the Lie algebra of the rotation group. The orthogonality and unit determinant of the matrix parameter are preserved using matrix logarithms and exponentials and the algorithm lends itself to interesting interpretations in terms of the computational topology of the compact Lie groups. The stability and the computational complexity of the algorithm are discussed.

A novel approach is presented for the detection and localization of changing periodicities in symbolic sequences. Various symbolic sequences like DNA can be modelled as stochastic processes that exhibit time-varying cyclostation- arity. The coding regions of the DNA, for instance, exhibit statistical periodicity with period three. The complexity-regularized maximum-likelihood estimates are developed in this paper for the statistical period of symbolic sequences. The changing periodicities along the sequence are discovered by using sliding windows. A cumulative sum test is also presented to detect the change points. The formulation in this paper avoids any kind of numerical mapping for the symbolic DNA sequences and does not impose any algebraic structure.

A novel group-theoretic method is presented for denoising a three-dimensional scene in isotropic noise. Images of the scene at varying depths are regarded as reference stochastic processes on the unit sphere to formulate Weiner-Hopf equations for estimating the image at any given depth. These comprise a set of coupled linear integral equations on the unit sphere and are solved using Peter-Weyl theory of Fourier transform on the rotation group. The computational complexity of this algorithm is reduced using bi-invariance of the image correlations with respect to the stabilizer subgroup of the rotation group.

A novel group theoretical method is proposed for autonomous navigation based on a spherical image camera. The environment of a robot is captured on a sphere. The three dimensional scenes at two different points in the space are related by a transformation from the special Euclidean motion group which is the semi-direct product of the rotation and the translation groups. The motion of the robot is recovered by iteratively estimating the rotation and the translation in an expectation-maximization fashion.

An adaptive oscillator is a system that can lock onto a time-varying input signal, synchronizing its output to both the frequency and phase of the input. A wavetable oscillator generates a periodic output by indexing into a lookup table that stores a single period of the waveform. An adaptive wavetable oscillator (AWO) combines these two ideas in a technique which separates the periodic output waveform from the parameters that control the adaptation of the frequency and phase of the waveform. The key issues in the design of AWOs are: the kind of oscillator to use, the class of admissible inputs, the shape of the wavetable, the control parameters, and the adaptive algorithm that adjusts the parameters. Wavetable oscillators can be applied to track the beat in MIDI signals, or, after an appropriate psycho-acoustical data reduction, to the tracking of audio signals. This paper examines these issues through analysis and simulation, focusing on conditions that achieve the desired entrainment between output and input. Sound examples demonstrate the application to beat tracking.