SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Equation (1) gives drift and corruption tolerance as additional performance measures: Since the regret explicitly depends on  $\Phi_T$  and  $\Lambda_T$  we can define the drift tolerance and corruption tolerance as performance measures of the algorithm. Drift (Corruption) tolerance is the largest  $\Phi_T$  (largest  $\Lambda_T$ ) for which we still guarantee sub-linear in  $T$  regret. Thus, if Equation (1) is satisfied by an algorithm, then the drift and corruption tolerance is  $\mathcal{O}(T^{1 - l})$  and  $\mathcal{O}(T^{1 - n})$  respectively. Thus, higher these tolerances, the better an algorithm is as they indicate that the algorithm's degradation with drifts and corruptions are more graceful. Our method is the only one to have non-zero drift tolerance (Table 2).
3 The clipped SGD Algorithm
In this section, we formally describe the algorithm in Algorithm 1 that achieves the desiderata. Algorithm 1 produces an estimate at time  $t$  is given by  $\theta_t \gets \mathcal{P}_{\Theta}(\theta_{t-1} - \eta_t \mathrm{clip}(\nabla \mathcal{L}(X_t, \theta_{t-1}, \lambda))$ , where  $\mathcal{P}_{\Theta}$  is the projection operator on to  $\Theta$  and  $\mathrm{clip}(x, c) \coloneqq \min(1, c / \| x \|) x, \forall x \in \mathbb{R}^d, c \geq 0$ .
Algorithm 1 Clipped-SGD [55]
1: Input:  $(\eta_t)_{t > 1}, \lambda > 0, \theta_0 \in \Theta, T$  time-horizon
2: for each time  $t = 1, 2, \dots, \mathbf{d}$
3: Receive sample  $X_{t}$
4: Output  $\theta_t \gets \mathcal{P}_{\Theta}(\theta_{t-1} - \eta_t \mathrm{clip}(\nabla \mathcal{L}(X_t, \theta_{t-1}), \lambda))$
5: end for
Intuition for why Algorithm 1 can achieve Equation (1): If there is no distribution shift or corruptions, clipped SGD with appropriate learning rate converges to the true parameter [55]. Now, if there are corruptions, clipping limits the impact on the overall regret. On the other hand, when a distribution shift occurs, the dynamics of Algorithm 1 is equivalent to restarting estimation under the new distribution, which will converge to the true parameter of this shifted distribution [55].
4 Regret bounds when the gradient vectors have sub-gaussian tails
Before giving the general result, we consider the special case of sub-gaussian distributions in this section. We do so because (i) the additional structure of sub-gaussian distributions enable us to prove stronger results in Corollaries 4.6 and 4.7 in the sequel, and (ii) the proof techniques from this special case enables us to build towards the proof of the general setting.
Definition 4.1 (Sub-gaussian random vectors, [57]). A random vector  $Y \in \mathbb{R}^d$  with co-variance matrix  $\Sigma$  is said to be sub-gaussian, if for all  $\lambda > 0$  and  $u \in \mathbb{R}^d$  with  $\| u \| = 1$ ,  $\mathbb{E}[e^{\lambda \langle Y, u \rangle}] \leq e^{\frac{\lambda^2 \nu_{max}(\Sigma)}{2}}$ , where  $\nu_{max}(\Sigma)$  is the highest eigen-value of  $\Sigma$ .
Assumption 4.2 (Sub-gaussian assumption with known upper bound). For every  $t$ , and  $\theta \in \Theta$ , the 0 mean random vector  $\nabla \mathcal{L}(Z, \theta) - \nabla \mathcal{R}_t(\theta)$ , where  $Z \sim \mathbb{P}_t$  is sub-gaussian with covariance matrix upper-bounded in the positive semi-definite sense by an known positive-semi-definite matrix  $\Sigma$ .
The following is the main result of this section.
Theorem 4.3 (Informal version of Theorem 18.1). Suppose Assumption 4.2 holds. For every  $\delta \in (0,1)$ , if Algorithm 1 is run with constant step-size  $\eta = \frac{1}{mT^{\alpha}}$  for  $\alpha \in [0,1]$  and clipping value  $\lambda \geq G + \sigma + \mathcal{O}\left(\sqrt{\nu_{max}(\Sigma)\ln\left(\frac{T}{\delta}\right)}\right)$ , then with probability at-least  $1 - \delta$ ,
$$
R e g _ {T} \leq \mathcal {O} \Bigg (\underbrace {\Phi_ {T} T ^ {\alpha} + \sqrt {\frac {T ^ {1 + \alpha} \Phi_ {T} \left(\sqrt {\nu_ {m a x} (\Sigma) \ln \left(\frac {T}{\delta}\right)} + \sigma\right)}{m}}} _ {R e g r e t f r o m d i s t r i b u t i o n s h i f t} +
$$
$$
\underbrace {T ^ {1 - \frac {\alpha}{2}}} _ {m} \left(\sqrt {\nu_ {m a x} (\Sigma) \ln \left(\frac {T}{\delta}\right)} + \sigma\right) + \underbrace {\Lambda_ {T} T ^ {\frac {\alpha}{2}} \sqrt {m \lambda \mathcal {D}}} _ {\text {R e g r e t f r o m c o r r u p t i o n s}}). \tag {2}
$$
Finite-sample estimation error
The main achievement in Theorem 4.3 is in explicitly identifying how the choice of  $\alpha$  trade-offs the regret contributions from distribution shift, finite sample error and corruptions.
Remark 4.4 (Corruption and drift tolerance). When Theorem 4.3 is instantiated with  $\alpha \in (0,1)$ , Equation (2) yields a drift tolerance of  $\mathcal{O}(T^{1 - \alpha})$  and corruption tolerance of  $\mathcal{O}(T^{1 - \frac{\alpha}{2}})$ . In particular, instantiating with  $\alpha = 2/3$  yields  $\mathcal{O}(T^{1/3})$  and  $\mathcal{O}(T^{2/3})$  corruption and drift tolerance respectively.
We now read off several corollaries from this result.
Corollary 4.5 (Finite diameter assumption can be relaxed in the absence of corruptions). Let  $\Lambda_T = 0$  almost-surely, i.e., there are no adversarial corruptions. Then, under the settings of Theorem 4.3 even when the set  $\Theta$  is unbounded, the regret bound given in Equation (2) holds.
Corollary 4.6 (Optimal in the stationary case): Under the assumptions of Theorem 4.3, if  $\Phi_T = \Lambda_T = 0$ , then when Algorithm 1 is run with parameters  $\lambda = +\infty$  and  $\eta = 1 / mT$ , the regret bound  $Reg_T \leq \mathcal{O}(\frac{\sqrt{T}}{m} (\sqrt{\nu_{max}(\Sigma)} \ln(T / \delta) + \sigma))$  holds with probability at-least  $1 - \delta$ .
This corollary recovers the well known result [24] of convergence of SGD on strongly convex functions with sub-gaussian gradients.
Corollary 4.7 (Optimal in the noiseless setting). Under the assumptions of Theorem 4.3, if  $\Lambda_T = 0$ , and  $\sigma = 0$ , i.e., there is no noise, then running Algorithm 1 with  $\lambda = +\infty$  and  $\eta = 1/m$  obtains regret bound of  $Reg_T = \mathcal{O}(\Phi_T)$ .
This result matches the lower bound in the noise-less setting [67] and recovers the upper-bound by previous analysis of online clipped-SGD specialized to the noiseless setting [43].
Corollary 4.8 (Special case of  $\alpha = 2/3$ ). When Theorem 4.3 is instantiated with  $\alpha = 2/3$ , then w.p. at-least  $1 - \delta$ , a regret bound of  $Reg_{T} \leq \mathcal{O}((T^{1/3}\Phi_{T} + \frac{T^{2/3}}{m} + T^{1/3}\Lambda_{T})poly(\sigma, G, \ln(T/\delta)))$ .
Remark 4.9 (The excess risk can be bounded using smoothness). Oftentimes, the regret is also measured through the excess risk, i.e.,  $\sum_{t=1}^{T} \mathbb{E}_{Z \sim \mathbb{P}_t}[\mathcal{L}(Z, \theta_t) - \mathcal{L}(Z, \theta_t^*)]$ . Since we assume that  $\mathcal{L}$  is  $M$  smooth (Assumption 2.1), Theorem 4.3 also bounds the excess risk regret by observing that  $\mathbb{E}_{Z \sim \mathbb{P}_t}[\mathcal{L}(Z, \theta_t) - \mathcal{L}(Z, \theta_t^*)] \leq \frac{M}{2} \| \theta_t - \theta_t^* \|^2$ .
4.1 Proof sketch for Theorem 4.3 and technical innovations
The full proof is Appendix 18. We first establish due to the condition on  $\lambda$  that if an input sample is not corrupted, then the gradient will never be clipped. Then in Lemma 19.6, we expand the one-step
recursion of the clipped SGD updates by exploiting the strong-convexity. In order to account for the effect of corruptions, we expand the recursion differently if a time-step is corrupted or not. If a time-step is corrupted, then we accumulate both a bias term of norm at-most  $\lambda$  and an appropriate variance term. If one were only interested in bounds in expectation, then the proof will follow the standard clipped SGD analysis as the expectation of the variance terms are 0.
To prove a high probability bound, a natural approach is to apply martingale concentrations to bound the variance terms as done in [24] in the absence of drifts and corruptions. A naive way to apply martingale concentrations would be to bound the variance error terms due to drifts and corruption by the worst case error by using the finite diameter  $\mathcal{D}$ . However, this will not lead to the dimension free bound of Theorem 4.3 — instead will lead to a bound in where the finite diameter  $\mathcal{D}$  will also multiply the regret term due to finite sample error. In particular, this approach will not result in Corollary 4.5 of being able to relax the finite diameter assumption in the absence of corruptions.
We circumvent this issue by using an inductive argument to bound the variance terms in Lemma 19.9. Concretely, we prove the induction hypothesis that for each time  $t$ , error terms until time  $t$  is bounded by an appropriate function of the drifts and corruptions. To establish the induction for a time  $t$ , we condition on the event the hypothesis holds till time  $t - 1$ , and employ martingale concentration to show the error at time  $t$  is small. Then we plug the martingale bound back into the one-step recursion and show the induction hypothesis also holds at time  $t$ . To get the final un-conditional result, we un-roll all the conditional events by an union bound.
5 Regret bounds in the general heavy-tailed case
The following is the main result of the paper, where we relax Assumption 4.2 and thus making the algorithm free of distributional knowledge and allowing for potentially heavy-tailed data.
Theorem 5.1 (Informal version of Theorem 20.1). When Algorithm 1 is run with clip value  $\lambda = 2GT^{\frac{3}{2}}$ , and step-size  $\eta = \frac{1}{mT^{\alpha}}$  for  $\alpha \in [0,1]$ , then for every  $\delta \in (0,1)$
$$
R e g _ {T} \leq \mathcal {O} \left(\underbrace {\frac {G \sigma T ^ {\frac {4 \alpha}{3}} \Phi_ {T}}{m ^ {3 / 2}} + \frac {\sigma T ^ {\frac {1}{2} + \frac {5 \alpha}{6}} \sqrt {\Phi_ {T}}}{m ^ {3 / 2} \sqrt {G}}} _ {\text {R e g r e t d u e t o d i s t r i b u t i o n s h i f t}} + \underbrace {\frac {T ^ {1 - \frac {\alpha}{3}} (G \sigma) ^ {2} \ln \left(\frac {T}{\delta}\right)}{m}} _ {\text {F i n i t e - s a m p l e e s t i m a t i o n e r r o r}} + \underbrace {\frac {T ^ {\frac {3 \alpha}{2}} \Lambda_ {T} G ^ {2} \sigma \sqrt {\mathcal {D}}}{m}} _ {\text {R e g r e t f r o m c o r r u p t i o n s}}\right), \tag {3}
$$
holds with probability at-least  $1 - \delta$
The achievement in Theorem 5.1 is in explicitly identifying how the choice of  $\alpha$  trade-offs the regret contributions from distribution shift, finite sample error and corruptions, despite heavy-tailed data.
Remark 5.2 (Price for relaxing assumption 4.2 is sub-optimal bound in the stationary case:). The setting in Theorem 5.1 is weaker compared to that of Theorem 4.3 since  $(i)$  no sub-gaussian tail assumptions are made, and  $(ii)$  no knowledge of the upper-bound matrix  $\Sigma$  is assumed. The price for relaxing these assumptions is a weaker regret, specifically in the term due to finite-sample estimation error. This term scales as  $\mathcal{O}(T^{1 - \frac{\alpha}{3}})$  in Theorem 5.1 while it scales as  $\mathcal{O}(T^{1 - \frac{\alpha}{2}})$  in Theorem 4.3. In particular in the stationary case, when Equation (3) is instantiated with  $\alpha = 1$ , the regret bound reads as  $\mathrm{Reg}_T = \mathcal{O}(T^{2/3}\sigma^2)$  which is weaker than the optimal  $\mathcal{O}(\sqrt{T}\sigma)$  for the stationary case [55, 39].
Corollary 5.3 (Setting  $\alpha = 1/2$ ). Under the conditions of Theorem 5.1, if Algorithm 1 is run with  $\alpha = 1/2$ , then with probability at-least  $1 - \delta$ , it satisfies a regret bound of  $\operatorname{Reg}_T \leq \mathcal{O}((T^{2/3}\Phi_T + T^{11/12}\sqrt{\Phi_T} + T^{5/6} + T^{3/4}\Lambda_T)\text{poly}(\sigma, G, 1/m, \ln(T/\delta)))$ .
Corollary 5.4 (Finite diameter assumption can be relaxed in the absence of corruptions). Let  $\Lambda_T = 0$  almost-surely, i.e., there are no adversarial corruptions. Then, under the settings of Theorem 5.1 even when the set  $\Theta$  is unbounded, the regret bound given in Equation (3) holds.
Remark 5.5 (Corruption and drift tolerance). Theorem 5.1 when instantiated with  $\alpha \in (0,1)$  yields a drift tolerance of  $\mathcal{O}(T^{1 - \frac{4\alpha}{3}})$  and corruption tolerance of  $\mathcal{O}(T^{1 - \frac{3\alpha}{2}})$ . In particular, instantiating with  $\alpha = 1/2$  yields  $\mathcal{O}(T^{1/3})$  and  $\mathcal{O}(T^{1/4})$  corruption and drift tolerance respectively.
(c)
Figure 2: Plot of regret versus  $\alpha$  over 10000 steps for mean-estimation. We observe that the best choice of  $\alpha$  decreases as  $\Phi_T$  and/or  $\Lambda_T$  increases. More details in Section 7.
5.1 Proof sketch of Theorem 5.1 and technical innovations
The proof for this case builds upon the techniques used to prove Theorem 4.3. Unlike in the sub-gaussian case, we cannot guarantee that only corrupted samples will be clipped in the general case. This introduces an additional bias and variance terms due to clipping of potential un-corrupted samples or inliers. The bias terms of the inliers are handled using techniques from prior works [25, 55]. We bound the variance terms using an inductive argument similar in spirit to that of Theorem 4.3. However, identifying the correct hypothesis involving drifts and outliers so that the induction argument holds is challenging. The challenge is because the induction hypothesis assumed till time  $t - 1$  impacts the martingale error term, which in-turn impacts whether the induction hypothesis will hold at time  $t$ . We introduce a free parameter to the regret term and deduce the induction hypothesis to hold if a quadratic equation in this parameter does not have any real roots. This extra term contributes to regret degradation compared to the sub-gaussian case (Lemma 20.10).
6 Insights and remarks
Known Time Horizon: This is a benign assumption and can be overcome by the standard doubling trick (Section 2.3 of [11]) with additional  $\log^2(T)$  factor regret (c.f. Appendix 14).
Price for being adaptive to distribution shift: Theorems 4.3 and 5.1 show that, in order to minimize regret due to statistical estimation, we need to set  $\alpha = 1$ , i.e., choose a learning rate of  $O(1 / T)$ . This was shown to be optimal in the absence of drift and corruptions both under sub-gaussian assumptions [24] and in heavy tail settings [55]. On the other hand, we see from Theorems 4.3 and 5.1, that in the presence of drift  $\Phi_T > 0$ , a learning rate  $\alpha < 1$  is sufficient to ensure sub-linear regret. The following lower bound (proved in Appendix 15) shows that  $\alpha < 1$  is also necessary.
Proposition 6.1 (Lower bound showing  $O(1 / T)$  learning rate is not adaptive to drifts). There exists a loss function  $\mathcal{L}$  such that for every  $T$ , there exists a sequence of probability measures  $(\mathbb{P}_t)_{t\geq}$  with the diameter  $\mathcal{D}\leq 2\log (T)$ , distribution shift  $\Phi_T\leq 2\log (T)$ ,  $\Lambda_T = 0$ , such that Algorithm 1 when run with  $\lambda \geq 1$  and step size  $\eta_t\coloneqq \frac{1}{t + 1}$  will incur regret at-least  $\frac{T}{3}$  with probability 1.
Finite diameter  $\mathcal{D}$  is necessary in general: We already saw in Proposition 2.6 that  $\Omega(\Lambda_T\mathcal{D})$  regret is necessary. We prove the lower-bound by considering mean-estimation in Section 16. Variants of Proposition 2.6 was observed in the literature (for ex. Proposition 1.3 [13], Lemma 6.1 in [33], Theorem D.1 in [14], Line 8 of Algorithm 3 in [19]). In theoretical statistics literature,  $\Omega(\Lambda_T\mathcal{D})$  is considered un-desirable [13] and thus the models studied restrict corruptions to occur at random times [13, 19] rather than arbitrary as in our setup. However, in practice corruptions are rarely random and typically occur close in time for instance due to machine failures or other external confounding factors [23, 42].
Regret bounds are dimension-free: The problem dimensions  $d, p$  do not appear in the regret bound. Moreover, the finite diameter  $\mathcal{D}$  only appears in the regret term affecting through adversarial corruptions and not in the distribution shift and finite sample error terms. Further, Corollaries 4.5 and 5.4 give regret bounds that hold even when  $\mathcal{D} = \infty$  in the absence of corruptions.
Theorems 4.3 and 5.1 give upper bounds on regret showing that tuning the learning rate depending on the amount of distribution shift and number of corruptions in the stream is beneficial. We empirically observe similar phenomena in Figures 2 and 3 (in the Appendix), thus indicating our theoretical observations are fundamental and not artifact of our bounds. We consider  $\mathcal{L}(X,\theta) = \frac{1}{2}\| X - \theta \| ^2$  corresponding to mean-estimation in Figure 2 and linear-regression of  $\mathcal{L}(X,\theta) = \frac{1}{2}\| X^{(2)} - \theta^T X^{(1)}\|^2$ , where  $X = (X^{(1)},X^{(2)})$  with  $X^{(1)}\in \mathbb{R}^{d - 1}$ ,  $X^{(2)}\in \mathbb{R}$ ,  $\theta \in \mathbb{R}^{(d - 1)}$  in Figure 3 in the Appendix in Section 12. We compare clipped-SGD with learning rate  $1 / mT^{\alpha}$  for variety of  $\alpha$  with clipping  $\lambda = 2T^{0.25}$  and use the doubling trick to remove dependence of  $T$  (i.e., use Algorithm 2). For the case of  $\alpha = 1$ , we use the learning rate of  $\eta_t = 1 / (m(t + 1))$  and  $\lambda = 2\sqrt{T}$  as suggested in [55]. All plots are averaged over 30 runs and we plot the median along with the 5th and 95th quantile bands. We observe in Figures 2 and 3 that as  $\Phi_T$  or  $\Lambda_T$  increases, the optimal  $\alpha$  to use decreases, validating the theoretical insight. Further details in the Appendix in Section 12. Evaluations on real-data is conducted in the Appendix in Section 13.
FITNESS for mean estimation is proposed in [50] that requires variance as input and is adaptive to drifts and corruptions under sub-gaussian distributions with per-sample computational complexity is  $O(dT)$ . In contrast, our algorithm applies to any strongly convex function, does not require moment bounds, data can be heavy-tailed. The work of [8] studied regret bounds in the absence of corruptions and only give bounds in expectation. A sequence of papers in online estimation including [44, 46, 13, 25, 55, 50] have derived algorithms that are robust in different ways. However, none of them consider the impact of distribution shift. The works of [46, 13] study robust linear regression in the absence of drifts with [46] making Gaussian assumptions on both covariates and response while [13] makes Gaussian assumption only on the responses. The work of [55] show that clipped SGD attains near optimal rates for any strongly convex loss function in the absence of drifts and corruptions. The paper of [19] study online estimation of strongly convex loss functions with corruptions, but do not consider drift. Moreover, their regret bounds are not dimension free (cf. Table 2). Although the paper of [64] gives regret bounds for online learning, a more general setting compared to estimation, do not consider impact of drifts. High probability bounds for optimization with heavy-tailed data have been studied by [44, 25, 15, 60, 36] in recent times. However, none of their analysis considers the impact of drift and corruptions. More related work in Section 10.
9 Conclusions and Open Problems
We studied online estimation in the presence of drifts and corruptions and potentially heavy-tailed inlier data and proved regret bounds for clipped SGD for strongly convex loss functions. Ours is the first proof that an online algorithm can simultaneously be robust to heavy-tails and corruptions and adaptive to drift. A key result was to show how the choice of learning-rate trades off drift, finite sample error and corruptions. Our work leaves several interesting open problems.
- The optimal choice of  $\alpha$  in Theorems 4.3 and 4.3 is a trade-off between distribution shift, finite sample error and corruptions. Can the optimal  $\alpha$  be set without knowing  $\Phi_T$ ,  $\Lambda_T$  or  $\sigma$  in advance?
- In the absence of corruptions, can regret  $\mathcal{O}((\sqrt{T} + T^a \Phi_T^{1 - a}) \mathrm{poly}(G, \sigma, \log(T / \delta))$ , for some  $a < 1$  be achieved? Such an algorithm would simultaneously have both  $(i)$ $\mathcal{O}(T)$  distribution shift tolerance which is the best possible and matching the lower bound established in [50], and  $(ii)$  closing the gap of Remark 5.2 for the stationary case.
- Can  $m = 1/2$  in Equation (1) be achieved in the general case with drift and corruptions? Theorem 5.1 shows that only  $m \geq 2/3$  is achievable. For the special case of stationary stream, [55] shows  $m = 1/2$  is achievable, matching the lower bound that  $m \geq 1/2$  is necessary.
- (Open problem from [55]) Does there exist an online algorithm that can obtain the statistically optimal regret of  $\mathcal{O}(\sqrt{T} (\sqrt{\mathrm{Tr}(\Sigma)} +\sqrt{\nu_{max}(\Sigma)\log(T / \delta)}))$  for a stationary stream?
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10 More related work