We study linear TD(0) under Markovian sampling, where data are generated along a single trajectory. We provide high-probability guarantees for a plain \emph{unprojected} TD(0) algorithm with Polyak–Ruppert (PR) averaging, using a \emph{single} stepsize schedule $\eta_t \propto 1/(\tau_{\mathrm{mix}}\log(t)\,\sqrt{t})$ that depends on mixing time but requires \emph{no prior knowledge of the curvature parameter $\omega$}. Our first result shows that such a choice of the stepsize guarantees that the TD(0) iterates are automatically and uniformly bounded \emph{with high probability}, without projections and without any stability argument based on $\omega$. Building on this result, we establish a simultaneous high-probability convergence guarantee for the PR average: the same stepsize yields both a robust curvature-free $\widetilde{\mathcal O}(\tau_{\mathrm{mix}}/\sqrt{T})$ rate and a fast curvature-dependent $\widetilde{\mathcal O}(\tau_{\mathrm{mix}}^2/(\omega T))$ rate, with the bound taking the minimum of the two. The core technical ingredient is a Poisson-equation toolkit for geometrically mixing Markov chains, which decomposes Markov noise into a martingale term plus a controlled remainder and enables a new self-bounding inductive argument for pathwise stability.

We study the problem of classifying the nodes of a given graph in the self-directed learning setup. This learning setting is a variant of online learning, where rather than an adversary determining the sequence in which nodes are presented, the learner autonomously and adaptively selects them. While self-directed learning of Euclidean halfspaces, linear functions, and general multiclass hypothesis classes was recently considered, no results previously existed specifically for self-directed node classification on graphs. In this paper, we address this problem developing efficient algorithms for it. More specifically, we focus on the case of (geodesically) convex clusters, i.e., for every two nodes sharing the same label, all nodes on every shortest path between them also share the same label. In particular, we devise an algorithm with runtime polynomial in $n$ that makes only $3(h(G)+1)^4 \ln n$ mistakes on graphs with two convex clusters, where $n$ is the total number of nodes and $h(G)$ is the Hadwiger number, i.e., the size of the largest clique minor of the graph $G$. We also show that our algorithm is robust to the case that clusters are slightly non-convex, still achieving a mistake bound logarithmic in $n$. Finally, we devise a simple and efficient algorithm for homophilic clusters, where strongly connected nodes tend to belong to the same class.

We study fundamental limits of first\hyp order stochastic optimization in a range of non\hyp convex settings, including $L$-smooth functions satisfying Quasar\hyp Convexity ({QC}), Quadratic Growth ({QG}), and Restricted Secant Inequalities ({RSI}). While the convergence properties of standard algorithms are well understood in deterministic regimes, significantly fewer results address the stochastic case, where only unbiased and noisy gradients are available. We establish new lower bounds on the number of noisy gradient queries to minimize these classes of functions, also showing that they are tight (up to a logarithmic factor) in all the relevant quantities characterizing each class. Our approach reformulates the optimization task as a function identification problem, leveraging \textit{divergence decomposition} arguments to construct a challenging subclass that leads to sharp lower bounds. Furthermore, we present a specialized algorithm in the one\hyp dimensional setting that achieves faster rates, suggesting that certain dimensional thresholds are intrinsic to the complexity of non\hyp convex stochastic optimization.

In this paper, we theoretically investigate the effects of noisy labels in offline alignment, with a focus on the interplay between privacy and robustness against adversarial corruption. Specifically, under linear modeling assumptions, we present a unified analysis covering both reinforcement learning from human feedback (RLHF) and direct preference optimization (DPO) under different privacy-corruption scenarios, such as Local differential privacy-then-Corruption (LTC), where human preference labels are privatized before being corrupted by an adversary, and Corruption-then-Local differential privacy (CTL), where labels are corrupted before privacy protection. Our analysis leverages a reduction framework that reduces the offline alignment problem under linear modeling assumptions to parameter estimation in logistic regression. This framework allows us to establish an interesting separation result between LTC and CTL, demonstrating that LTC presents a greater challenge than CTL in offline alignment, even under linear models. As important by-products, our findings also advance the state-of-the-art theoretical results in offline alignment under privacy-only or corruption-only scenarios.

In this paper, we theoretically study the offline alignment of language models with human preference feedback, under both preference label corruption and privacy protections. To this end, we propose a variant of $\chi$PO – Square$\chi$PO, which is a simple one-line change of $\chi$PO with the standard log-loss replaced by a new square loss over probability. Thanks to the inherent nice properties of this new loss, we have advanced the state-of-the-art of differentially private and robust alignment. Specifically, for the local model of label privacy, Square$\chi$PO is the first one that attains optimal rate based on single-policy concentrability even with general function approximations. It also gives the first result under the central model of privacy protection over both prompts (responses) and labels. On the robustness side against Huber label corruption, Square$\chi$PO is the first alignment method that has a meaningful theoretical guarantee under general function approximations. More importantly, Square$\chi$PO can address privacy protection and corruption simultaneously, where an interesting separation is observed, implying that the order of privacy and corruption matters. Furthermore, we show that Square$\chi$PO can also be easily extended to handle the scenario of the general preference model with state-of-the-art guarantees under corruption and privacy. Last but not least, all of our theoretical guarantees enjoy a unified analysis, building upon a new result on the generalization error bounds of least-square regression under corruption and privacy constraints, which we believe is of independent interest to the community.

Asynchronous methods are fundamental for parallelizing computations in distributed machine learning. They aim to accelerate training by fully utilizing all available resources. However, their greedy approach can lead to inefficiencies using more computation than required, especially when computation times vary across devices. If the computation times were known in advance, training could be fast and resource-efficient by assigning more tasks to faster workers. The challenge lies in achieving this optimal allocation without prior knowledge of the computation time distributions. In this paper, we propose ATA (Adaptive Task Allocation), a method that adapts to heterogeneous and random distributions of worker computation times. Through rigorous theoretical analysis, we show that ATA identifies the optimal task allocation and performs comparably to methods with prior knowledge of computation times. Experimental results further demonstrate that ATA is resource-efficient, significantly reducing costs compared to the greedy approach, which can be arbitrarily expensive depending on the number of workers.

As sequential learning algorithms are increasingly applied to real life, ensuring data privacy while maintaining their utilities emerges as a timely question. In this context, regret minimisation in stochastic bandits under $\epsilon$-global Differential Privacy (DP) has been widely studied. The present literature poses a significant gap between the best-known regret lower and upper bound in this setting, though they ``match in order''. Thus, we revisit the regret lower and upper bounds of $\epsilon$-global DP bandits and improve both. First, we prove a tighter regret lower bound involving a novel information-theoretic quantity characterising the hardness of $\epsilon$-global DP in stochastic bandits. This quantity smoothly interpolates between Kullback–Leibler divergence and Total Variation distance, depending on the privacy budget $\epsilon$. Then, we choose two asymptotically optimal bandit algorithms, i.e., KL-UCB and IMED, and propose their DP versions using a unified blueprint, i.e., (a) running in arm-dependent phases, and (b) adding Laplace noise to achieve privacy. For Bernoulli bandits, we analyse the regrets of these algorithms and show that their regrets asymptotically match our lower bound up to a constant arbitrary close to 1. At the core of our algorithms lies a new concentration inequality for sums of Bernoulli variables under Laplace mechanism, which is a new DP version of the Chernoff bound. Finally, our numerical experiments validate that DP-KLUCB and DP-IMED achieve lower regret than the existing $\epsilon$-global DP bandit algorithms.

We study dynamic regret in online convex optimization, where the objective is to achieve low cumulative loss relative to an arbitrary benchmark sequence. By observing that competing with an arbitrary sequence of comparators $u_{1},\ldots,u_{T}$ in $\mathcal{W}\subseteq\mathbb{R}^{d}$ can be reframed as competing with a *fixed* comparator *function* $u:[1,T]\to \mathcal{W}$, we cast dynamic regret minimization as a *static regret* problem in a *function space*. By carefully constructing a suitable function space in the form of a Reproducing Kernel Hilbert Space (RKHS), our reduction enables us to recover the optimal $R_{T}(u_{1},\ldots,u_{T}) = \mathcal{O}(\sqrt{\sum_{t}\\|u_{t}-u_{t-1}\\|T})$ dynamic regret guarantee in the setting of linear losses, and yields new scale-free and directionally-adaptive dynamic regret guarantees. Moreover, unlike prior dynamic-to-static reductions---which are valid only for linear losses---our reduction holds for *any* sequence of losses, allowing us to recover $\mathcal{O}\big(\\|u\\|^2_{\mathcal{H}}+d_{\mathrm{eff}}(\lambda)\ln T\big)$ bounds when the losses have meaningful curvature, where $d_{\mathrm{eff}}(\lambda)$ is a measure of complexity of the RKHS. Despite working in an infinite-dimensional space, the resulting reduction leads to algorithms that are computable in practice, due to the reproducing property of RKHSs.

The Polyak stepsize has been proven to be a fundamental stepsize in convex optimization, giving near optimal gradient descent rates across a wide range of assumptions. The universality of the Polyak stepsize has also inspired many stochastic variants, with theoretical guarantees and strong empirical performance. Despite the many theoretical results, our understanding of the convergence properties and shortcomings of the Polyak stepsize or its variants is both incomplete and fractured across different analyses. We propose a new, unified, and simple perspective for the Polyak stepsize and its variants as gradient descent on a surrogate loss. We show that each variant is equivalent to minimize a surrogate function with stepsizes that adapt to a guaranteed local curvature. Our general surrogate loss perspective is then used to provide a unified analysis of existing variants across different assumptions. Moreover, we show a number of negative results proving that the non-convergence results in some of the upper bounds is indeed real.

Science progresses by iteratively advancing and correcting humanity's understanding of the world. In machine learning (ML) research, rapid advancements have led to an explosion of publications, but have also led to misleading, incorrect, flawed or perhaps even fraudulent studies being accepted and sometimes highlighted at ML conferences due to the fallibility of peer review. While such mistakes are understandable, ML conferences do not offer robust processes to help the field systematically correct when such errors are made.This position paper argues that ML conferences should establish a dedicated "Refutations and Critiques" (R&C) Track. This R&C Track would provide a high-profile, reputable platform to support vital research that critically challenges prior research, thereby fostering a dynamic self-correcting research ecosystem.We discuss key considerations including track design, review principles, potential pitfalls, and provide an illustrative example submission concerning a recent ICLR 2025 Oral.We conclude that ML conferences should create official, reputable mechanisms to help ML research self-correct.

The construction of confidence intervals for the mean of a bounded random variable is a classical problem in statistics with numerous applications in machine learning and virtually all scientific fields. In particular, obtaining the tightest possible confidence intervals is vital every time the sampling of the random variables is expensive. The current state-of-the-art method to construct confidence intervals is by using betting algorithms. This is a very successful approach for deriving optimal confidence sequences, even matching the rate of law of iterated logarithms. However, in the fixed horizon setting, these approaches are either sub-optimal or based on heuristic solutions with strong empirical performance but without a finite-time guarantee. Hence, no betting-based algorithm guaranteeing the optimal $\mathcal{O}(\sqrt{\frac{\sigma^2\log\frac1\delta}{n}})$ width of the confidence intervals are known. This work bridges this gap. We propose a betting-based algorithm to compute confidence intervals that empirically outperforms the competitors. Our betting strategy uses the optimal strategy in every step (in a certain sense), whereas the standard betting methods choose a constant strategy in advance. Leveraging this fact results in strict improvements even for classical concentration inequalities, such as the ones of Hoeffding or Bernstein. Moreover, we also prove that the width of our confidence intervals is optimal up to an $1+o(1)$ factor diminishing with $n$.

Let $f(\theta, X_1),$ $ …,$ $ f(\theta, X_n)$ be a sequence of random elements, where $f$ is a fixed scalar function, $X_1, …, X_n$ are independent random variables (data), and $\theta$ is a random parameter distributed according to some data-dependent \emph{posterior} distribution $P_n$. In this paper, we consider the problem of proving concentration inequalities to estimate the mean of the sequence. An example of such a problem is the estimation of the generalization error of some predictor trained by a stochastic algorithm, such as a neural network, where $f$ is a loss function. Classically, this problem is approached through a \emph{PAC-Bayes} analysis where, in addition to the posterior, we choose a \emph{prior} distribution which captures our belief about the inductive bias of the learning problem. Then, the key quantity in PAC-Bayes concentration bounds is a divergence that captures the \emph{complexity} of the learning problem where the de facto standard choice is the Kullback-Leibler (KL) divergence. However, the tightness of this choice has rarely been questioned. In this paper, we challenge the tightness of the KL-divergence-based bounds by showing that it is possible to achieve a strictly tighter bound. In particular, we demonstrate new \emph{high-probability} PAC-Bayes bounds with a novel and \emph{better-than-KL} divergence that is inspired by Zhang et al. (2022). Our proof is inspired by recent advances in regret analysis of gambling algorithms, and its use to derive concentration inequalities. Our result is first-of-its-kind in that existing PAC-Bayes bounds with non-KL divergences are not known to be strictly better than KL. Thus, we believe our work marks the first step towards identifying optimal rates of PAC-Bayes bounds.

Normalization layers are one of the key building blocks for deep neural networks. Several theoretical studies have shown that batch normalization improves the signal propagation, by avoiding the representations from becoming collinear across the layers. However, results on mean-field theory of batch normalization also conclude that this benefit comes at the expense of exploding gradients in depth. Motivated by these two aspects of batch normalization, in this study we pose the following question: *Can a batch-normalized network keep the optimal signal propagation properties, but avoid exploding gradients?* We answer this question in the affirmative by giving a particular construction of an *MLP with linear activations* and batch-normalization that provably has *bounded gradients* at any depth. Based on Weingarten calculus, we develop a rigorous and non-asymptotic theory for this constructed MLP that gives a precise characterization of forward signal propagation, while proving that gradients remain bounded for linearly independent input samples, which holds in most practical settings. Inspired by our theory, we also design an activation shaping scheme that empirically achieves the same properties for non-linear activations.

We study the problem of dynamic regret minimization in online convex optimization, in which the objective is to minimize the difference between the cumulative loss of an algorithm and that of an arbitrary sequence of comparators. While the literature on this topic is very rich, a unifying framework for the analysis and design of these algorithms is still missing. In this paper we show that *for linear losses, dynamic regret minimization is equivalent to static regret minimization in an extended decision space*. Using this simple observation, we show that there is a frontier of lower bounds trading off penalties due to the variance of the losses and penalties due to variability of the comparator sequence, and provide a framework for achieving any of the guarantees along this frontier. As a result, we also prove for the first time that adapting to the squared path-length of an arbitrary sequence of comparators to achieve regret $R_{T}(u_{1},\dots,u_{T})\le O(\sqrt{T\sum_{t} \\|u_{t}-u_{t+1}\\|^{2}})$ is impossible. However, using our framework we introduce an alternative notion of variability based on a locally-smoothed comparator sequence $\bar u_{1}, \dots, \bar u_{T}$, and provide an algorithm guaranteeing dynamic regret of the form $R_{T}(u_{1},\dots,u_{T})\le \tilde O(\sqrt{T\sum_{i}\\|\bar u_{i}-\bar u_{i+1}\\|^{2}})$, while still matching in the worst case the usual path-length dependencies up to polylogarithmic terms.

A classic problem in statistics is the estimation of the expectation of random variables from samples. This gives rise to the tightly connected problems of deriving concentration inequalities and confidence sequences, i.e., confidence intervals that hold uniformly over time. Previous studies have shown that it is possible to convert the regret guarantee of an online learning algorithm into concentration inequalities, but these concentration results were not tight. In this paper, we show regret guarantees of universal portfolio algorithms applied to the online learning problem of betting give rise to new implicit time-uniform concentration inequalities for bounded random variables. The key feature of our concentration results is that they are centered around the maximum log wealth of the best fixed betting strategy in hindsight. We propose numerical methods to solve these implicit inequalities, which results in confidence sequences that enjoy the empirical Bernstein rate with the optimal asymptotic behavior while being never worse than Bernoulli-KL confidence bounds. We further show that our confidence sequences are never vacuous with even one sample, for any given target failure rate δ ∈ (0,1). Our empirical study shows that our confidence bounds achieve the state-of-the-art performance, especially in the small sample regime.

We consider the problem of estimating the mean of a sequence of random elements $f(\theta, X_1)$ $, \ldots, $ $f(\theta, X_n)$ where $f$ is a fixed scalar function, $S=(X_1, \ldots, X_n)$ are independent random variables, and $\theta$ is a possibly $S$-dependent parameter. An example of such a problem would be to estimate the generalization error of a neural network trained on $n$ examples where $f$ is a loss function. Classically, this problem is approached through concentration inequalities holding uniformly over compact parameter sets of functions $f$, for example as in Rademacher or VC type analysis. However, in many problems, such inequalities often yield numerically vacuous estimates. Recently, the \emph{PAC-Bayes} framework has been proposed as a better alternative for this class of problems for its ability to often give numerically non-vacuous bounds. In this paper, we show that we can do even better: we show how to refine the proof strategy of the PAC-Bayes bounds and achieve \emph{even tighter} guarantees. Our approach is based on the \emph{coin-betting} framework that derives the numerically tightest known time-uniform concentration inequalities from the regret guarantees of online gambling algorithms. In particular, we derive the first PAC-Bayes concentration inequality based on the coin-betting approach that holds simultaneously for all sample sizes. We demonstrate its tightness showing that by \emph{relaxing} it we obtain a number of previous results in a closed form including Bernoulli-KL and empirical Bernstein inequalities. Finally, we propose an efficient algorithm to numerically calculate confidence sequences from our bound, which often generates nonvacuous confidence bounds even with one sample, unlike the state-of-the-art PAC-Bayes bounds.

We propose a new class of online learning algorithms, generalized implicit Follow-The-Regularized-Leader (FTRL), that expands the scope of FTRL framework. Generalized implicit FTRL can recover known algorithms, such as FTRL with linearized losses and implicit FTRL, and it allows the design of new update rules, as extensions of aProx and Mirror-Prox to FTRL. Our theory is constructive in the sense that it provides a simple unifying framework to design updates that directly improve the worst-case upper bound on the regret. The key idea is substituting the linearization of the losses with a Fenchel-Young inequality. We show the flexibility of the framework by proving that some known algorithms, like the Mirror-Prox updates, are instantiations of the generalized implicit FTRL. Finally, the new framework allows us to recover the temporal variation bound of implicit OMD, with the same computational complexity.

We present new algorithms for optimizing non-smooth, non-convex stochastic objectives based on a novel analysis technique. This improves the current best-known complexity for finding a $(\delta,\epsilon)$-stationary point from $O(\epsilon^{-4}\delta^{-1})$ stochastic gradient queries to $O(\epsilon^{-3}\delta^{-1})$, which we also show to be optimal. Our primary technique is a reduction from non-smooth non-convex optimization to *online learning*, after which our results follow from standard regret bounds in online learning. For *deterministic and second-order smooth* objectives, applying more advanced optimistic online learning techniques enables a new complexity of $O(\epsilon^{-1.5}\delta^{-0.5})$. Our improved non-smooth analysis also immediately recovers all optimal or best-known results for finding $\epsilon$ stationary points of smooth or second-order smooth objectives in both stochastic and deterministic settings.

Parameter-free stochastic gradient descent (PFSGD) algorithms do not require setting learning rates while achieving optimal theoretical performance. In practical applications, however, there remains an empirical gap between tuned stochastic gradient descent (SGD) and PFSGD. In this paper, we close the empirical gap with a new parameter-free algorithm based on continuous-time Coin-Betting on truncated models. The new update is derived through the solution of an Ordinary Differential Equation (ODE) and solved in a closed form. We show empirically that this new parameter-free algorithm outperforms algorithms with the ``best default'' learning rates and almost matches the performance of finely tuned baselines without anything to tune.

Parameter-free algorithms are online learning algorithms that do not require setting learning rates. They achieve optimal regret with respect to the distance between the initial point and any competitor. Yet, parameter-free algorithms do not take into account the geometry of the losses. Recently, in the stochastic optimization literature, it has been proposed to instead use truncated linear lower bounds, which produce better performance by more closely modeling the losses. In particular, truncated linear models greatly reduce the problem of overshooting the minimum of the loss function. Unfortunately, truncated linear models cannot be used with parameter-free algorithms because the updates become very expensive to compute. In this paper, we propose new parameter-free algorithms that can take advantage of truncated linear models through a new update that has an “implicit” flavor. Based on a novel decomposition of the regret, the new update is efficient, requires only one gradient at each step, never overshoots the minimum of the truncated model, and retains the favorable parameter-free properties. We also conduct an empirical study demonstrating the practical utility of our algorithms.

SGD with Momentum (SGDM) is a widely used family of algorithms for large scale optimization of machine learning problems. Yet, when optimizing generic convex functions, no advantage is known for any SGDM algorithm over plain SGD. Moreover, even the most recent results require changes to the SGDM algorithms, like averaging of the iterates and a projection onto a bounded domain, which are rarely used in practice. In this paper, we focus on the convergence rate of the last iterate of SGDM. For the first time, we prove that for any constant momentum factor, there exists a Lipschitz and convex function for which the last iterate of SGDM suffers from a suboptimal convergence rate of $\Omega(\frac{\log T}{\sqrt{T}})$ after $T$ iterations. Based on this fact, we study a class of (both adaptive and non-adaptive) Follow-The-Regularized-Leader-based SGDM algorithms with \emph{increasing momentum} and \emph{shrinking updates}. For these algorithms, we show that the last iterate has optimal convergence $O(\frac{1}{\sqrt{T}})$ for unconstrained convex stochastic optimization problems without projections onto bounded domains nor knowledge of $T$. Further, we show a variety of results for FTRL-based SGDM when used with adaptive stepsizes. Empirical results are shown as well.

Convex-concave min-max problems are ubiquitous in machine learning, and people usually utilize first-order methods (e.g., gradient descent ascent) to find the optimal solution. One feature which separates convex-concave min-max problems from convex minimization problems is that the best known convergence rates for min-max problems have an explicit dependence on the size of the domain, rather than on the distance between initial point and the optimal solution. This means that the convergence speed does not have any improvement even if the algorithm starts from the optimal solution, and hence, is oblivious to the initialization. Here, we show that strict-convexity-strict-concavity is sufficient to get the convergence rate to depend on the initialization. We also show how different algorithms can asymptotically achieve initialization-dependent convergence rates on this class of functions. Furthermore, we show that the so-called “parameter-free” algorithms allow to achieve improved initialization-dependent asymptotic rates without any learning rate to tune. In addition, we utilize this particular parameter-free algorithm as a subroutine to design a new algorithm, which achieves a novel non-asymptotic fast rate for strictly-convex-strictly-concave min-max problems with a growth condition and H{ö}lder continuous solution mapping. Experiments are conducted to verify our theoretical findings and demonstrate the effectiveness of the proposed algorithms.

Traditional analyses in non-convex optimization typically rely on the smoothness assumption, namely requiring the gradients to be Lipschitz. However, recent evidence shows that this smoothness condition does not capture the properties of some deep learning objective functions, including the ones involving Recurrent Neural Networks and LSTMs. Instead, they satisfy a much more relaxed condition, with potentially unbounded smoothness. Under this relaxed assumption, it has been theoretically and empirically shown that the gradient-clipped SGD has an advantage over the vanilla one. In this paper, we show that clipping is not indispensable for Adam-type algorithms in tackling such scenarios: we theoretically prove that a generalized SignSGD algorithm can obtain similar convergence rates as SGD with clipping but does not need explicit clipping at all. This family of algorithms on one end recovers SignSGD and on the other end closely resembles the popular Adam algorithm. Our analysis underlines the critical role that momentum plays in analyzing SignSGD-type and Adam-type algorithms: it not only reduces the effects of noise, thus removing the need for large mini-batch in previous analyses of SignSGD-type algorithms, but it also substantially reduces the effects of unbounded smoothness and gradient norms. To the best of our knowledge, this work is the first one showing the benefit of Adam-type algorithms compared with non-adaptive gradient algorithms such as gradient descent in the unbounded smoothness setting. We also compare these algorithms with popular optimizers on a set of deep learning tasks, observing that we can match the performance of Adam while beating others.

Inspired by the demands of real-time climate and weather forecasting, we develop optimistic online learning algorithms that require no parameter tuning and have optimal regret guarantees under delayed feedback. Our algorithms—DORM, DORM+, and AdaHedgeD—arise from a novel reduction of delayed online learning to optimistic online learning that reveals how optimistic hints can mitigate the regret penalty caused by delay. We pair this delay-as-optimism perspective with a new analysis of optimistic learning that exposes its robustness to hinting errors and a new meta-algorithm for learning effective hinting strategies in the presence of delay. We conclude by benchmarking our algorithms on four subseasonal climate forecasting tasks, demonstrating low regret relative to state-of-the-art forecasting models.

Stochastic Gradient Descent (SGD) is a popular tool in training large-scale machine learning models. Its performance, however, is highly variable, depending crucially on the choice of the step sizes. Accordingly, a variety of strategies for tuning the step sizes have been proposed, ranging from coordinate-wise approaches (a.k.a. “adaptive” step sizes) to sophisticated heuristics to change the step size in each iteration. In this paper, we study two step size schedules whose power has been repeatedly confirmed in practice: the exponential and the cosine step sizes. For the first time, we provide theoretical support for them proving convergence rates for smooth non-convex functions, with and without the Polyak-Ł{}ojasiewicz (PL) condition. Moreover, we show the surprising property that these two strategies are \emph{adaptive} to the noise level in the stochastic gradients of PL functions. That is, contrary to polynomial step sizes, they achieve almost optimal performance without needing to know the noise level nor tuning their hyperparameters based on it. Finally, we conduct a fair and comprehensive empirical evaluation of real-world datasets with deep learning architectures. Results show that, even if only requiring at most two hyperparameters to tune, these two strategies best or match the performance of various finely-tuned state-of-the-art strategies.

Quantile (and, more generally, KL) regret bounds, such as those achieved by NormalHedge (Chaudhuri, Freund, and Hsu 2009) and its variants, relax the goal of competing against the best individual expert to only competing against a majority of experts on adversarial data. More recently, the semi-adversarial paradigm (Bilodeau, Negrea, and Roy 2020) provides an alternative relaxation of adversarial online learning by considering data that may be neither fully adversarial nor stochastic (I.I.D.). We achieve the minimax optimal regret in both paradigms using FTRL with separate, novel, root-logarithmic regularizers, both of which can be interpreted as yielding variants of NormalHedge. We extend existing KL regret upper bounds, which hold uniformly over target distributions, to possibly uncountable expert classes with arbitrary priors; provide the first full-information lower bounds for quantile regret on finite expert classes (which are tight); and provide an adaptively minimax optimal algorithm for the semi-adversarial paradigm that adapts to the true, unknown constraint faster, leading to uniformly improved regret bounds over existing methods.

In the setting of online learning, Implicit algorithms turn out to be highly successful from a practical standpoint. However, the tightest regret analyses only show marginal improvements over Online Mirror Descent. In this work, we shed light on this behavior carrying out a careful regret analysis. We prove a novel static regret bound that depends on the temporal variability of the sequence of loss functions, a quantity which is often encountered when considering dynamic competitors. We show, for example, that the regret can be constant if the temporal variability is constant and the learning rate is tuned appropriately, without the need of smooth losses. Moreover, we present an adaptive algorithm that achieves this regret bound without prior knowledge of the temporal variability and prove a matching lower bound. Finally, we validate our theoretical findings on classification and regression datasets.

Stochastic gradient descent is the method of choice for large scale optimization of machine learning objective functions. Yet, its performance is greatly variable and heavily depends on the choice of the stepsizes. This has motivated a large body of research on adaptive stepsizes. However, there is currently a gap in our theoretical understanding of these methods, especially in the non-convex setting. In this paper, we start closing this gap: we theoretically analyze in the convex and non-convex settings a generalized version of the AdaGrad stepsizes. We show sufficient conditions for these stepsizes to achieve almost sure asymptotic convergence of the gradients to zero, proving the first guarantee for generalized AdaGrad stepsizes in the non-convex setting. Moreover, we show that these stepsizes allow to automatically adapt to the level of noise of the stochastic gradients in both the convex and non-convex settings, interpolating between O(1/T) and O(1/sqrt(T)), up to logarithmic terms.

We consider the problem of unconstrained online convex optimization (OCO) with sub-exponential noise, a strictly more general problem than the standard OCO. In this setting, the learner receives a subgradient of the loss functions corrupted by sub-exponential noise and strives to achieve optimal regret guarantee, without knowledge of the competitor norm, i.e., in a parameter-free way. Recently, Cutkosky and Boahen (COLT 2017) proved that, given unbounded subgradients, it is impossible to guarantee a sublinear regret due to an exponential penalty. This paper shows that it is possible to go around the lower bound by allowing the observed subgradients to be unbounded via stochastic noise. However, the presence of unbounded noise in unconstrained OCO is challenging; existing algorithms do not provide near-optimal regret bounds or fail to have a guarantee. So, we design a novel parameter-free OCO algorithm for Banach space, which we call BANCO, via a reduction to betting on noisy coins. We show that BANCO achieves the optimal regret rate in our problem. Finally, we show the application of our results to obtain a parameter-free locally private stochastic subgradient descent algorithm, and the connection to the law of iterated logarithms.

Stochastic Gradient Descent (SGD) has played a central role in machine learning. However, it requires a carefully hand-picked stepsize for fast convergence, which is notoriously tedious and time-consuming to tune. Over the last several years, a plethora of adaptive gradient-based algorithms have emerged to ameliorate this problem. In this paper, we propose new surrogate losses to cast the problem of learning the optimal stepsizes for the stochastic optimization of a non-convex smooth objective function onto an online convex optimization problem. This allows the use of no-regret online algorithms to compute optimal stepsizes on the fly. In turn, this results in a SGD algorithm with self-tuned stepsizes that guarantees convergence rates that are automatically adaptive to the level of noise.

Variance reduction has emerged in recent years as a strong competitor to stochastic gradient descent in non-convex problems, providing the first algorithms to improve upon the converge rate of stochastic gradient descent for finding first-order critical points. However, variance reduction techniques typically require carefully tuned learning rates and willingness to use excessively large "mega-batches" in order to achieve their improved results. We present a new algorithm, STORM, that does not require any batches and makes use of adaptive learning rates, enabling simpler implementation and less hyperparameter tuning. Our technique for removing the batches uses a variant of momentum to achieve variance reduction in non-convex optimization. On smooth losses $F$, STORM finds a point $x$ with $\mathbb{E}[\|\nabla F(x)\|]\le O(1/\sqrt{T}+\sigma^{1/3}/T^{1/3})$ in $T$ iterations with $\sigma^2$ variance in the gradients, matching the best-known rate but without requiring knowledge of $\sigma$.

In this paper we consider the nonparametric least square regression in a Reproducing Kernel Hilbert Space (RKHS). We propose a new randomized algorithm that has optimal generalization error bounds with respect to the square loss, closing a long-standing gap between upper and lower bounds. Moreover, we show that our algorithm has faster finite-time and asymptotic rates on problems where the Bayes risk with respect to the square loss is small. We state our results using standard tools from the theory of least square regression in RKHSs, namely, the decay of the eigenvalues of the associated integral operator and the complexity of the optimal predictor measured through the integral operator.

This paper presents a new approach, called perturb-max, for high-dimensional statistical inference in graphical models that is based on applying random perturbations followed by optimization. This framework injects randomness into maximum a-posteriori (MAP) predictors by randomly perturbing the potential function for the input. A classic result from extreme value statistics asserts that perturb-max operations generate unbiased samples from the Gibbs distribution using high-dimensional perturbations. Unfortunately, the computational cost of generating so many high-dimensional random variables can be prohibitive. However, when the perturbations are of low dimension, sampling the perturb-max prediction is as efficient as MAP optimization. This paper shows that the expected value of perturb-max inference with low dimensional perturbations can be used sequentially to generate unbiased samples from the Gibbs distribution. Furthermore the expected value of the maximal perturbations is a natural bound on the entropy of such perturb-max models. A measure concentration result for perturb-max values shows that the deviation of their sampled average from its expectation decays exponentially in the number of samples, allowing effective approximation of the expectation.

We introduce several new black-box reductions that significantly improve the design of adaptive and parameter-free online learning algorithms by simplifying analysis, improving regret guarantees, and sometimes even improving runtime. We reduce parameter-free online learning to online exp-concave optimization, we reduce optimization in a Banach space to one-dimensional optimization, and we reduce optimization over a constrained domain to unconstrained optimization. All of our reductions run as fast as online gradient descent. We use our new techniques to improve upon the previously best regret bounds for parameter-free learning, and do so for arbitrary norms.

This paper describes a new parameter-free online learning algorithm for changing environments. In comparing against algorithms with the same time complexity as ours, we obtain a strongly adaptive regret bound that is a factor of at least $\sqrt\log(T)$ better, where $T$ is the time horizon. Empirical results show that our algorithm outperforms state-of-the-art methods in learning with expert advice and metric learning scenarios.

Deep learning methods achieve state-of-the-art performance in many application scenarios. Yet, these methods require a significant amount of hyperparameters tuning in order to achieve the best results. In particular, tuning the learning rates in the stochastic optimization process is still one of the main bottlenecks. In this paper, we propose a new stochastic gradient descent procedure for deep networks that does not require any learning rate setting. Contrary to previous methods, we do not adapt the learning rates nor we make use of the assumed curvature of the objective function. Instead, we reduce the optimization process to a game of betting on a coin and propose a learning rate free optimal algorithm for this scenario. Theoretical convergence is proven for convex and quasi-convex functions and empirical evidence shows the advantage of our algorithm over popular stochastic gradient algorithms.

Existing vanilla algorithms for online linear optimization have O((ηR(u) + 1/η) \sqrtT) regret with respect to any competitor u, where R(u) is a 1-strongly convex regularizer and η> 0 is a tuning parameter of the algorithm. For certain decision sets and regularizers, the so-called \emphparameter-free algorithms have \widetilde O(\sqrtR(u) T) regret with respect to any competitor u. Vanilla algorithm can achieve the same bound only for a fixed competitor u known ahead of time by setting η= 1/\sqrtR(u). A drawback of both vanilla and parameter-free algorithms is that they assume that the norm of the loss vectors is bounded by a constant known to the algorithm. There exist \emphscale-free algorithms that have O((ηR(u) + 1/η) \sqrtT \max_1 \le t \le T \norm\ell_t) regret with respect to any competitor u and for any sequence of loss vector \ell_1, …, \ell_T. Parameter-free analogue of scale-free algorithms have never been designed. Is is possible to design algorithms that are simultaneously \emphparameter-free and \emphscale-free?

We develop a novel preconditioning method for ridge regression, based on recent linear sketching methods. By equipping Stochastic Variance Reduced Gradient (SVRG) with this preconditioning process, we obtain a significant speed-up relative to fast stochastic methods such as SVRG, SDCA and SAG.

In the recent years, a number of parameter-free algorithms have been developed for online linear optimization over Hilbert spaces and for learning with expert advice. These algorithms achieve optimal regret bounds that depend on the unknown competitors, without having to tune the learning rates with oracle choices. We present a new intuitive framework to design parameter-free algorithms for both online linear optimization over Hilbert spaces and for learning with expert advice, based on reductions to betting on outcomes of adversarial coins. We instantiate it using a betting algorithm based on the Krichevsky-Trofimov estimator. The resulting algorithms are simple, with no parameters to be tuned, and they improve or match previous results in terms of regret guarantee and per-round complexity.

Stream mining poses unique challenges to machine learning: predictive models are required to be scalable, incrementally trainable, must remain bounded in size, and benon parametric in order to achieve high accuracy even in complex and dynamic environments. Moreover, the learning system must be parameterless - traditional tuning methods are problematic in streaming settings - and avoid requiring prior knowledge of the number of distinct class labels occurring in the stream. In this paper, we introduce a new algorithmic approach for nonparametric learning in data streams. Our approach addresses all above mentioned challenges by learning a model that covers the input space using simple local classifiers. The distribution of these classifiers dynamically adapts to the local (unknown) complexity of the classification problem, thus achieving a good balance between model complexity and predictive accuracy. By means of an extensive empirical evaluation against standard nonparametric baselines, we show state-of-the-art results in terms of accuracy versus model size. Our empirical analysis is complemented by a theoretical performance guarantee which does not rely on any stochastic assumption on the source generating the stream.

We study algorithms for online linear optimization in Hilbert spaces, focusing on the case where the player is unconstrained. We develop a novel characterization of a large class of minimax algorithms, recovering, and even improving, several previous results as immediate corollaries. Moreover, using our tools, we develop an algorithm that provides a regret bound of O(U \sqrtT \log( U \sqrtT \log^2 T +1)), where U is the L_2 norm of an arbitrary comparator and both T and U are unknown to the player. This bound is optimal up to \sqrt\log \log T terms. When T is known, we derive an algorithm with an optimal regret bound (up to constant factors). For both the known and unknown T case, a Normal approximation to the conditional value of the game proves to be the key analysis tool.

The maximum a-posteriori (MAP) perturbation framework has emerged as a useful approach for inference and learning in high dimensional complex models. By maximizing a randomly perturbed potential function, MAP perturbations generate unbiased samples from the Gibbs distribution. Unfortunately, the computational cost of generating so many high-dimensional random variables can be prohibitive. More efficient algorithms use sequential sampling strategies based on the expected value of low dimensional MAP perturbations. This paper develops new measure concentration inequalities that bound the number of samples needed to estimate such expected values. Applying the general result to MAP perturbations can yield a more efficient algorithm to approximate sampling from the Gibbs distribution. The measure concentration result is of general interest and may be applicable to other areas involving Monte Carlo estimation of expectations.

Learning a visual object category from few samples is a compelling and challenging problem. In several real-world applications collecting many annotated data is costly and not always possible. However, a small training set does not allow to cover the high intraclass variability typical of visual objects. In this condition, machine learning methods provide very few guarantees. This paper presents a discriminative model adaptation algorithm able to proficiently learn a target object with few examples by relying on other previously learned source categories. The proposed method autonomously chooses from where and how much to transfer information by solving a convex optimization problem which ensures to have the minimal leave-one-out error on the available training set. We analyze several properties of the described approach and perform an extensive experimental comparison with other existing transfer solutions, consistently showing the value of our algorithm.

Kernelized Support Vector Machines (SVM) have gained the status of off-the-shelf classifiers, able to deliver state of the art performance on almost any problem. Still, their practical use is constrained by their computational and memory complexity, which grows super-linearly with the number of training samples. In order to retain the low training and testing complexity of linear classifiers and the flexibility of non linear ones, a growing, promising alternative is represented by methods that learn non-linear classifiers through local combinations of linear ones. In this paper we propose a new multi class local classifier, based on a latent SVM formulation. The proposed classifier makes use of a set of linear models that are linearly combined using sample and class specific weights. Thanks to the latent formulation, the combination coefficients are modeled as latent variables. We allow soft combinations and we provide a closed-form solution for their estimation, resulting in an efficient prediction rule. This novel formulation allows to learn in a principled way the sample specific weights and the linear classifiers, in a unique optimization problem, using a CCCP optimization procedure. Extensive experiments on ten standard UCI machine learning datasets, one large binary dataset, three character and digit recognition databases, and a visual place categorization dataset show the power of the proposed approach.

Since the seminal work of Thrun [16], the learning to learn paradigm has been defined as the ability of an agent to improve its performance at each task with experience, with the number of tasks. Within the object categorization domain, the visual learning community has actively declined this paradigm in the transfer learning setting. Almost all proposed methods focus on category detection problems, addressing how to learn a new target class from few samples by leveraging over the known source. But if one thinks of learning over multiple tasks, there is a need for multiclass transfer learning algorithms able to exploit previous source knowledge when learning a new class, while at the same time optimizing their overall performance. This is an open challenge for existing transfer learning algorithms. The contribution of this paper is a discriminative method that addresses this issue, based on a Least-Squares Support Vector Machine formulation. Our approach is designed to balance between transferring to the new class and preserving what has already been learned on the source models. Extensive experiments on subsets of publicly available datasets prove the effectiveness of our approach.

We consider the transfer learning scenario, where the learner does not have access to the source domain directly, but rather operates on the basis of hypotheses induced from it – the Hypothesis Transfer Learning (HTL) problem. Particularly, we conduct a theoretical analysis of HTL by considering the algorithmic stability of a class of HTL algorithms based on Regularized Least Squares with biased regularization. We show that the relatedness of source and target domains accelerates the convergence of the Leave-One-Out error to the generalization error, thus enabling the use of the Leave-One-Out error to find the optimal transfer parameters, even in the presence of a small training set. In case of unrelated domains we also suggest a theoretically principled way to prevent negative transfer, so that in the limit we recover the performance of the algorithm not using any knowledge from the source domain.

We consider the problem of maintaining the data-structures of a partition-based regression procedure in a setting where the training data arrives sequentially over time. We prove that it is possible to maintain such a structure in time $O(\log n)$ at any time step $n$ while achieving a nearly-optimal regression rate of $\tilde{O}(n^{-2/(2+d)})$ in terms of the unknown metric dimension $d$. Finally we prove a new regression lower-bound which is independent of a given data size, and hence is more appropriate for the streaming setting.

We present a new online learning algorithm that extends the exponentiated gradient to infinite dimensional spaces. Our analysis shows that the algorithm is implicitly able to estimate the $L_2$ norm of the unknown competitor, $U$, achieving a regret bound of the order of $O(U \log (U T+1))\sqrt{T})$, instead of the standard $O((U^2 +1) \sqrt{T})$, achievable without knowing $U$. For this analysis, we introduce novel tools for algorithms with time-varying regularizers, through the use of local smoothness. Through a lower bound, we also show that the algorithm is optimal up to $\sqrt{\log T}$ term for linear and Lipschitz losses.

At the time of this writing, the main means of control for polyarticulated self-powered hand prostheses is surface electromyography (sEMG). In the clinical setting, data collected from two electrodes are used to guide the hand movements selecting among a finite number of postures. Machine learning has been applied in the past to the sEMG signal (not in the clinical setting) with interesting results, which provide more insight on how these data could be used to improve prosthetic functionality. Researchers have mainly concentrated so far on increasing the accuracy of sEMG classification and/or regression, but, in general, a finer control implies a longer training period. A desirable characteristic would be to shorten the time needed by a patient to learn how to use the prosthesis. To this aim, we propose here a general method to reuse past experience, in the form of models synthesized from previous subjects, to boost the adaptivity of the prosthesis. Extensive tests on databases recorded from healthy subjects in controlled and noncontrolled conditions reveal that the method significantly improves the results over the baseline nonadaptive case. This promising approach might be employed to pretrain a prosthesis before shipping it to a patient, leading to a shorter training phase.

Open ended learning is a dynamic process based on the continuous analysis of new data, guided by past experience. On one side it is helpful to take advantage of prior knowledge when only few information on a new task is available (transfer learning). On the other, it is important to continuously update an existing model so to exploit the new incoming data, especially if their informative content is very different from what is already known (online learning). Until today these two aspects of the learning process have been tackled separately. In this paper we propose an algorithm that takes the best of both worlds: we consider a sequential learning setting, and we exploit the potentiality of knowledge transfer with a computationally cheap solution. At the same time, by relying on past experience we boost online learning to predict reliably on future problems. A theoretical analysis, coupled with extensive experiments, show that our approach performs well in terms of the online number of training mistakes, as well as in terms of performance on separate test sets.

We present a novel multilabel/ranking algorithm working in partial information settings. The algorithm is based on 2nd-order descent methods, and relies on upper-confidence bounds to trade-off exploration and exploitation. We analyze this algorithm in a partial adversarial setting, where covariates can be adversarial, but multilabel probabilities are ruled by (generalized) linear models. We show $O(T^{1/2}\log T)$ regret bounds, which improve in several ways on the existing results. We test the effectiveness of our upper-confidence scheme by contrasting against full-information baselines on real-world multilabel datasets, often obtaining comparable performance.

Several object categorization algorithms use kernel methods over multiple cues, as they offer a principled approach to combine multiple cues, and to obtain state-of-the-art performance. A general drawback of these strategies is the high computational cost during training, that prevents their application to large-scale problems. They also do not provide theoretical guarantees on their convergence rate. Here we present a Multiclass Multi Kernel Learning (MKL) algorithm that obtains state-of-the-art performance in a considerably lower training time. We generalize the standard MKL formulation to introduce a parameter that allows us to decide the level of sparsity of the solution. Thanks to this new setting, we can directly solve the problem in the primal formulation. We prove theoretically and experimentally that 1) our algorithm has a faster convergence rate as the number of kernels grow; 2) the training complexity is linear in the number of training examples; 3) very few iterations are enough to reach good solutions. Experiments on three standard benchmark databases support our claims.

Learning object categories from small samples is a challenging problem, where machine learning tools can in general provide very few guarantees. Exploiting prior knowledge may be useful to reproduce the human capability of recognizing objects even from only one single view. This paper presents an SVM-based model adaptation algorithm able to select and weight appropriately prior knowledge coming from different categories. The method relies on the solution of a convex optimization problem which ensures to have the minimal leave-one-out error on the training set. Experiments on a subset of the Caltech-256 database show that the proposed method produces better results than both choosing one single prior model, and transferring from all previous experience in a flat uninformative way.

In many real world applications we do not have access to fully-labeled training data, but only to a list of possible labels. This is the case, e.g., when learning visual classifiers from images downloaded from the web, using just their text captions or tags as learning oracles. In general, these problems can be very difficult. However most of the time there exist different implicit sources of information, coming from the relations between instances and labels, which are usually dismissed. In this paper, we propose a semi-supervised framework to model this kind of problems. Each training sample is a bag containing multi-instances, associated with a set of candidate labeling vectors. Each labeling vector encodes the possible labels for the instances in the bag, with only one being fully correct. The use of the labeling vectors provides a principled way not to exclude any information. We propose a large margin discriminative formulation, and an efficient algorithm to solve it. Experiments conducted on artificial datasets and a real-world images and captions dataset show that our approach achieves performance comparable to SVM trained with the ground-truth labels, and outperforms other baselines.

We propose a general framework to online learning for classification problems with time-varying potential functions in the adversarial setting. This framework allows to design and prove relative mistake bounds for any generic loss function. The mistake bounds can be specialized for the hinge loss, allowing to recover and improve the bounds of known online classification algorithms. By optimizing the general bound we derive a new online classification algorithm, called NAROW, that hybridly uses adaptive- and fixed- second order information. We analyze the properties of the algorithm and illustrate its performance using synthetic dataset.

We introduce a new algorithm for binary classification in the selective sampling protocol. Our algorithm uses Regularized Least Squares (RLS) as base classifier, and for this reason it can be efficiently run in any RKHS. Unlike previous margin-based semi-supervised algorithms, our sampling condition hinges on a simultaneous upper bound on bias and variance of the RLS estimate under a simple linear label noise model. This fact allows us to prove performance bounds that hold for an arbitrary sequence of instances. In particular, we show that our sampling strategy approximates the margin of the Bayes optimal classifier to any desired accuracy ε by asking Õ (d/ε2) queries (in the RKHS case d is replaced by a suitable spectral quantity). While these are the standard rates in the fully supervised i.i.d. case, the best previously known result in our harder setting was Õ (d3/ε4). Preliminary experiments show that some of our algorithms also exhibit a good practical performance.

The state-of-the-art in control of hand prosthetics is far from optimal. The main control interface is represented by surface electromyography (EMG): the activation potentials of the remnants of large muscles of the stump are used in a non-natural way to control one or, at best, two degrees-of-freedom. This has two drawbacks: first, the dexterity of the prosthesis is limited, leading to poor interaction with the environment; second, the patient undergoes a long training time. As more dexterous hand prostheses are put on the market, the need for a finer and more natural control arises. Machine learning can be employed to this end. A desired feature is that of providing a pre-trained model to the patient, so that a quicker and better interaction can be obtained. To this end we propose model adaptation with least-squares SVMs, a technique that allows the automatic tuning of the degree of adaptation. We test the effectiveness of the approach on a database of EMG signals gathered from human subjects. We show that, when pre-trained models are used, the number of training samples needed to reach a certain performance is reduced, and the overall performance is increased, compared to what would be achieved by starting from scratch.

To fulfill the dream of having autonomous robots at home, there is a need for spatial representations augmented with semantic concepts. Vision has emerged recently as the key modality to recognize semantic categories like places (office, corridor, kitchen, etc). A crucial aspect of these semantic place representations is that they change over time, due to the dynamism of the world. This calls for visual algorithms able to learn from experience while at the same time managing the continuous flow of incoming data. This paper addresses these issues by presenting an SVM-based algorithm able to (a) learn continuously from experience with a fast updating rule, and (b) control the memory growth via a random forgetting mechanism while at the same time preserving an accuracy comparable to that of the batch algorithm. We apply our method to two different scenarios where learning from experience plays an important role: (1) continuous learning of visual places under dynamic changes, and (2) knowledge transfer of visual concepts across robot platforms. For both scenarios, results confirm the effectiveness of our approach.

We present a discriminative online algorithm with a bounded memory growth, which is based on the kernel-based Perceptron. Generally, the required memory of the kernel-based Perceptron for storing the online hypothesis is not bounded. Previous work has been focused on discarding part of the instances in order to keep the memory bounded. In the proposed algorithm the instances are not discarded, but projected onto the space spanned by the previous online hypothesis. We derive a relative mistake bound and compare our algorithm both analytically and empirically to the state-of-the-art Forgetron algorithm (Dekel et al, 2007). The first variant of our algorithm, called Projectron, outperforms the Forgetron. The second variant, called Projectron++, outperforms even the Perceptron.

In the framework of indoor mobile robotics, place recognition is a challenging task, where it is crucial that self-localization be enforced precisely, notwithstanding the changing conditions of illumination, objects being shifted around and/or people affecting the appearance of the scene. In this scenario online learning seems the main way out, thanks to the possibility of adapting to changes in a smart and flexible way. Nevertheless, standard machine learning approaches usually suffer when confronted with massive amounts of data and when asked to work online. Online learning requires a high training and testing speed, all the more in place recognition, where a continuous flow of data comes from one or more cameras. In this paper we follow the Support Vector Machines-based approach of Pronobis et al., proposing an improvement that we call Online Independent Support Vector Machines. This technique exploits linear independence in the image feature space to incrementally keep the size of the learning machine remarkably small while retaining the accuracy of a standard machine. Since the training and testing time crucially depend on the size of the machine, this solves the above stated problems. Our experimental results prove the effectiveness of the approach.

We consider the problem of estimating the relative orientation of a number of individual photocells-or pixels-that hold fixed relative positions. The photocells measure the intensity of on a pencil of lines. We assume that the light-field thus sampled is changing, e.g. as the result of motion of the sensors and use the obtained measurements to estimate the orientations of the photocells. We explore an information-theoretic and geometric approach: based on real-world data, we build anon-parametric functional relation linking the information distance between the data streams of two photocells, and the angular separation between the photocells. Then, given data streams produced by arrays of pixels in similar conditions, we use the functional relation to estimate the angles between pixels. Finally, we embed the estimated angles in the unit 3D sphere to obtain the estimated layout of the array.