Better Rates for Random Task Orderings in Continual Linear Models

We study the common continual learning setup where an overparameterized model is sequentially fitted to a set of jointly realizable tasks. We analyze the forgetting, i.e., loss on previously seen tasks, after $k$ iterations. For linear models, we prove that fitting a task is equivalent to a single stochastic gradient descent (SGD) step on a modified objective. We develop novel last-iterate SGD upper bounds in the realizable least squares setup, and apply them to derive new results for continual learning. Focusing on random orderings over $T$ tasks, we establish universal forgetting rates, whereas existing rates depend on the problem dimensionality or complexity. Specifically, in continual regression with replacement, we improve the best existing rate from $O((d-r)/k)$ to $O(\min(k^{-1/4}, \sqrt{d-r}/k, \sqrt{Tr}/k))$, where $d$ is the dimensionality and $r$ the average task rank. Furthermore, we establish the first rates for random task orderings without replacement. The obtained rate of $O(\min(T^{-1/4}, (d-r)/T))$ proves for the first time that randomization alone, with no task repetition, can prevent catastrophic forgetting in sufficiently long task sequences. Finally, we prove a similar $O(k^{-1/4})$ universal rate for the forgetting in continual linear classification on separable data. Our universal rates apply for broader projection methods, such as block Kaczmarz and POCS, illuminating their loss convergence under i.i.d and one-pass orderings.

Benefits of Learning Rate Annealing for Tuning-Robustness in Stochastic Optimization

The learning rate in stochastic gradient methods is a critical hyperparameter that is notoriously costly to tune via standard grid search, especially for training modern large-scale models with billions of parameters. We identify a theoretical advantage of learning rate annealing schemes that decay the learning rate to zero at a polynomial rate, such as the widely-used cosine schedule, by demonstrating their increased robustness to initial parameter misspecification due to a coarse grid search. We present an analysis in a stochastic convex optimization setup demonstrating that the convergence rate of stochastic gradient descent with annealed schedules depends sublinearly on the multiplicative misspecification factor $\rho$ (i.e., the grid resolution), achieving a rate of $O(\rho^{1/(2p+1)}/\sqrt{T})$ where $p$ is the degree of polynomial decay and $T$ is the number of steps, in contrast to the $O(\rho/\sqrt{T})$ rate that arises with fixed stepsizes and exhibits a linear dependence on $\rho$. Experiments confirm the increased robustness compared to tuning with a fixed stepsize, that has significant implications for the computational overhead of hyperparameter search in practical training scenarios.

Bandit Multiclass List Classification

We study the problem of multiclass list classification with (semi-)bandit feedback, where input examples are mapped into subsets of size $m$ of a collection of $K$ possible labels, and the feedback consists of the predicted labels which lie in the set of true labels of the given example. Our main result is for the $(\varepsilon,\delta)$-PAC variant of the problem for which we design an algorithm that returns an $\varepsilon$-optimal hypothesis with high probability using a sample complexity of $O \big( (\mathrm{poly}(K/m) + sm / \varepsilon^2) \log (|H|/\delta) \big)$ where $H$ is the underlying (finite) hypothesis class and $s$ is an upper bound on the number of true labels for a given example. This bound improves upon known bounds for combinatorial semi-bandits whenever $s \ll K$. Moreover, in the regime where $s = O(1)$ the leading terms in our bound match the corresponding full-information rates, implying that bandit feedback essentially comes at no cost. Our PAC learning algorithm is also computationally efficient given access to an ERM oracle for $H$. Additionally, we consider the regret minimization setting where data can be generated adversarially, and establish a regret bound of $\widetilde O(|H| + \sqrt{smT \log |H|})$. Our results generalize and extend those of Erez et al. (2024) who consider the simpler single-label setting corresponding to $s=m=1$, and in fact hold for the more general contextual combinatorial semi-bandit problem with $s$-sparse rewards.

Complexity of Vector-valued Prediction: From Linear Models to Stochastic Convex Optimization

We study the problem of learning vector-valued linear predictors: these are prediction rules parameterized by a matrix that maps an $m$-dimensional feature vector to a $k$-dimensional target. We focus on the fundamental case with a convex and Lipschitz loss function, and show several new theoretical results that shed light on the complexity of this problem and its connection to related learning models. First, we give a tight characterization of the sample complexity of Empirical Risk Minimization (ERM) in this setting, establishing that $\smash{\widetilde{\Omega}}(k/\epsilon^2)$ examples are necessary for ERM to reach $\epsilon$ excess (population) risk; this provides for an exponential improvement over recent results by Magen and Shamir (2023) in terms of the dependence on the target dimension $k$, and matches a classical upper bound due to Maurer (2016). Second, we present a black-box conversion from general $d$-dimensional Stochastic Convex Optimization (SCO) to vector-valued linear prediction, showing that any SCO problem can be embedded as a prediction problem with $k=\Theta(d)$ outputs. These results portray the setting of vector-valued linear prediction as bridging between two extensively studied yet disparate learning models: linear models (corresponds to $k=1$) and general $d$-dimensional SCO (with $k=\Theta(d)$).

Faster Stochastic Optimization with Arbitrary Delays via Asynchronous Mini-Batching

We consider the problem of asynchronous stochastic optimization, where an optimization algorithm makes updates based on stale stochastic gradients of the objective that are subject to an arbitrary (possibly adversarial) sequence of delays. We present a procedure which, for any given $q \in (0,1]$, transforms any standard stochastic first-order method to an asynchronous method with convergence guarantee depending on the $q$-quantile delay of the sequence. This approach leads to convergence rates of the form $O(\tau_q/qT+\sigma/\sqrt{qT})$ for non-convex and $O(\tau_q^2/(q T)^2+\sigma/\sqrt{qT})$ for convex smooth problems, where $\tau_q$ is the $q$-quantile delay, generalizing and improving on existing results that depend on the average delay. We further show a method that automatically adapts to all quantiles simultaneously, without any prior knowledge of the delays, achieving convergence rates of the form $O(\inf_{q} \tau_q/qT+\sigma/\sqrt{qT})$ for non-convex and $O(\inf_{q} \tau_q^2/(q T)^2+\sigma/\sqrt{qT})$ for convex smooth problems. Our technique is based on asynchronous mini-batching with a careful batch-size selection and filtering of stale gradients.

Fast Rates for Bandit PAC Multiclass Classification

We study multiclass PAC learning with bandit feedback, where inputs are classified into one of $K$ possible labels and feedback is limited to whether or not the predicted labels are correct. Our main contribution is in designing a novel learning algorithm for the agnostic $(\varepsilon,\delta)$-PAC version of the problem, with sample complexity of $O\big( (\operatorname{poly}(K) + 1 / \varepsilon^2) \log (|H| / \delta) \big)$ for any finite hypothesis class $H$. In terms of the leading dependence on $\varepsilon$, this improves upon existing bounds for the problem, that are of the form $O(K/\varepsilon^2)$. We also provide an extension of this result to general classes and establish similar sample complexity bounds in which $\log |H|$ is replaced by the Natarajan dimension. This matches the optimal rate in the full-information version of the problem and resolves an open question studied by Daniely, Sabato, Ben-David, and Shalev-Shwartz (2011) who demonstrated that the multiplicative price of bandit feedback in realizable PAC learning is $\Theta(K)$. We complement this by revealing a stark contrast with the agnostic case, where the price of bandit feedback is only $O(1)$ as $\varepsilon \to 0$. Our algorithm utilizes a stochastic optimization technique to minimize a log-barrier potential based on Frank-Wolfe updates for computing a low-variance exploration distribution over the hypotheses, and is made computationally efficient provided access to an ERM oracle over $H$.

Private Online Learning via Lazy Algorithms

We study the problem of private online learning, specifically, online prediction from experts (OPE) and online convex optimization (OCO). We propose a new transformation that transforms lazy online learning algorithms into private algorithms. We apply our transformation for differentially private OPE and OCO using existing lazy algorithms for these problems. Our final algorithms obtain regret, which significantly improves the regret in the high privacy regime $\varepsilon \ll 1$, obtaining $\sqrt{T \log d} + T^{1/3} \log(d)/\varepsilon^{2/3}$ for DP-OPE and $\sqrt{T} + T^{1/3} \sqrt{d}/\varepsilon^{2/3}$ for DP-OCO. We also complement our results with a lower bound for DP-OPE, showing that these rates are optimal for a natural family of low-switching private algorithms.

The Real Price of Bandit Information in Multiclass Classification

We revisit the classical problem of multiclass classification with bandit feedback (Kakade, Shalev-Shwartz and Tewari, 2008), where each input classifies to one of $K$ possible labels and feedback is restricted to whether the predicted label is correct or not. Our primary inquiry is with regard to the dependency on the number of labels $K$, and whether $T$-step regret bounds in this setting can be improved beyond the $\smash{\sqrt{KT}}$ dependence exhibited by existing algorithms. Our main contribution is in showing that the minimax regret of bandit multiclass is in fact more nuanced, and is of the form $\smash{\widetilde{\Theta}\left(\min \left\{|\mathcal{H}| + \sqrt{T}, \sqrt{KT \log |{\mathcal{H}|}} \right\} \right) }$, where $\mathcal{H}$ is the underlying (finite) hypothesis class. In particular, we present a new bandit classification algorithm that guarantees regret $\smash{\widetilde{O}(|\mathcal{H}|+\sqrt{T})}$, improving over classical algorithms for moderately-sized hypothesis classes, and give a matching lower bound establishing tightness of the upper bounds (up to log-factors) in all parameter regimes.

A Note on High-Probability Analysis of Algorithms with Exponential, Sub-Gaussian, and General Light Tails

This short note describes a simple technique for analyzing probabilistic algorithms that rely on a light-tailed (but not necessarily bounded) source of randomization. We show that the analysis of such an algorithm can be reduced, in a black-box manner and with only a small loss in logarithmic factors, to an analysis of a simpler variant of the same algorithm that uses bounded random variables and often easier to analyze. This approach simultaneously applies to any light-tailed randomization, including exponential, sub-Gaussian, and more general fast-decaying distributions, without needing to appeal to specialized concentration inequalities. Analyses of a generalized Azuma inequality and stochastic optimization with general light-tailed noise are provided to illustrate the technique.

How Free is Parameter-Free Stochastic Optimization?

We study the problem of parameter-free stochastic optimization, inquiring whether, and under what conditions, do fully parameter-free methods exist: these are methods that achieve convergence rates competitive with optimally tuned methods, without requiring significant knowledge of the true problem parameters. Existing parameter-free methods can only be considered ``partially'' parameter-free, as they require some non-trivial knowledge of the true problem parameters, such as a bound on the stochastic gradient norms, a bound on the distance to a minimizer, etc. In the non-convex setting, we demonstrate that a simple hyperparameter search technique results in a fully parameter-free method that outperforms more sophisticated state-of-the-art algorithms. We also provide a similar result in the convex setting with access to noisy function values under mild noise assumptions. Finally, assuming only access to stochastic gradients, we establish a lower bound that renders fully parameter-free stochastic convex optimization infeasible, and provide a method which is (partially) parameter-free up to the limit indicated by our lower bound.