SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Balakrishnan (Murali) Narayanaswamy
{abisanka, muralibn}@amazon.com, Amazon Web Services, Santa Clara CA, USA.
The real-time estimation of time-varying parameters from high-dimensional, heavily-tailed and corrupted data-streams is a common sub-routine in systems ranging from those for network monitoring and anomaly detection to those for traffic scheduling in data-centers. For estimation tasks that can be cast as minimizing a strongly convex loss function, we prove that an appropriately tuned version of the clipped Stochastic Gradient Descent (SGD) is simultaneously  $(i)$  adaptive to drift,  $(ii)$  robust to heavy-tailed inliers and arbitrary corruptions,  $(iii)$  requires no distributional knowledge and  $(iv)$  can be implemented in an online streaming fashion. All prior estimation algorithms have only been proven to possess a subset of these practical desiderata. A observation we make is that, neither the  $\mathcal{O}\left(\frac{1}{t}\right)$  learning rate for clipped SGD known to be optimal for strongly convex loss functions of a stationary data-stream, nor the  $\mathcal{O}(1)$  learning rate known to be optimal for being adaptive to drift in a noiseless environment can be used. Instead, a learning rate of  $T^{-\alpha}$  for  $\alpha < 1$  where  $T$  is the stream-length is needed to balance adaptivity to potential drift and to combat noise. We develop a new inductive argument and combine it with a martingale concentration result to derive high-probability under any learning rate on data-streams exhibiting arbitrary distribution shift - a proof strategy that may be of independent interest. Further, using the classical doubling-trick, we relax the knowledge of the stream length  $T$ . Ours is the first online estimation algorithm that is provably robust to heavy-tails, corruptions and distribution shift simultaneously. We complement our theoretical results empirically on synthetic and real data.
Technology improvements have made it easier than ever to collect diverse telemetry at high resolution from any cyber or physical system, for both monitoring and control [31]. This in turn has led to a data deluge, where large amounts of data must be processed at scale [7, 16]. Given the scale and velocity of the data sources, offline processing to make predictions and decisions is computationally cumbersome. For real-time applications such as performance monitoring and anomaly detection, offline/batch processing results in stale predictions [6], [61], [54]. This necessitates computationally cheap, online algorithms that make predictions and decisions on high-dimensional streaming data [41], [21], [58]. Further in many applications, the challenge of being restricted to online algorithms is exacerbated by heavy-tails [2], [3], distribution shift [48], [28] and outliers/anomalies in the data-stream [29], [42], [35, 50]. In practice, although several heuristics to circumvent these issues have been designed [1, 66, 48, 28, 35, 42], there is no systematic study of the impact of these challenges on the achievable statistical accuracy. Motivated by these problems, we study algorithms for high-dimensional online statistical estimation and rigorously establish performance guarantees which exhibit graceful degradation with distribution shift and outliers in the data-stream.
(b)
Figure 1: Figures  $(a)$  and  $(b)$  show a stream of independent samples (with some corruptions) from unit-variance Gaussian and Pareto distribution with shape parameter 2.001 respectively when the underlying true mean changes with time. Figure  $(c)$  shows that clipped-SGD with clip-value of 5 incurs lower regret in estimating the true time-varying mean when the learning rate is set to  $1 / \sqrt{T}$  as opposed to the standard  $1 / (t + 1)$  known to be optimal in a stationary environment.
1.1 Problem Setup
The online estimation problem we study is modeled as an interaction between an oblivious adversary and the estimator. The adversary before the interaction begins makes two choices - (i) a sequence of probability distributions  $\mathbb{P}_1, \dots, \mathbb{P}_T$  on  $\mathbb{R}^d$  and (ii) a sequence of measurable corruption functions  $(\mathcal{C}_t)_t = 1$  where for every  $t \in [T]^1$ ,  $\mathcal{C}_t: \mathbb{R}^{d \times t} \times \mathbb{R}^{d \times t-1} \times \mathbb{R}^{p \times t-1} \to \mathbb{R}^d$ , whose roles we explain below.
Sequential interaction protocol: At each time  $t \in [T]$ , the vector  $X_{t} \coloneqq Z_{t} + C_{t}$ , where  $Z_{t} \sim \mathbb{P}_{t}$  is sampled independently and  $C_{t} \coloneqq \mathcal{C}_{t}((Z_{s})_{s \leq t}, (C_{s})_{s < t}, (\theta_{s})_{s < t})$  is the corruption vector computed using the corruption function based on the past random samples and outputs.  $X_{t}$  is shown as input to the estimation algorithm. Subsequently after observing  $X_{t}$ , the estimation algorithm outputs an estimate  $\theta_{t} \in \Theta$  and incurs loss  $l_{t} \coloneqq \| \theta_{t} - \theta_{t}^{*} \|^{2}$ . Here,  $\mathcal{L}: \mathbb{R}^{d} \times \mathbb{R}^{p} \to \mathbb{R}$  is a strongly convex loss function,  $\Theta \subset \mathbb{R}^{p}$  a closed convex set denoting the parameter space, and  $\theta_{t}^{*} \coloneqq \arg \min_{\theta \in \Theta} \mathbb{E}_{Z \sim \mathbb{P}_{t}}[\mathcal{L}(Z, \theta)]$  is the unique<sup>3</sup> optimizer of the loss function with respect to the distribution  $\mathbb{P}_{t}$ , chosen by the adversary at time  $t$ . For every time  $t$ , we denote this minimizer  $\theta_{t}^{*} \in \Theta$  as the true parameter at time  $t$ . The incurred loss  $l_{t}$  is un-observed by the algorithm since the true parameter  $\theta_{t}^{*}$  is unknown.
Formally, an estimation $^4$  algorithm  $\mathfrak{A} \coloneqq (\mathcal{A}_t)_t = 1$  is a sequence of measurable functions such that for all  $t \in [T]$ , the estimate output  $\theta_t \coloneqq \mathcal{A}_t(X_t, \dots, X_1)$ , where  $\mathcal{A}_t: \mathbb{R}^{d \times t} \to \mathbb{R}^p$  is a function of all inputs up to and including time  $t$ . After outputting an estimate  $\theta_t$ , the algorithm incurs an un-observed loss which is the L2 distance between the estimate and the true parameter.
Causal property of the adversary: Recall that at each time  $t$ , the corruption vector is given by  $C_t = \mathcal{C}_t((Z_s)_{s \leq t}, (C_s)_{s < t}, (\theta_s)_{s < t})$ , which is based only on the past realizations  $Z_1, \dots, Z_t$ ,  $C_1, \dots, C_{t-1}$  and the estimator's past outputs  $\theta_1, \dots, \theta_{t-1}$ . The adversary does not have access to future randomness  $Z_{t+1}, \dots, Z_T$  while choosing the corruption vector  $C_t$ . This model of corruptions is stronger than those used in online robust estimation (cf. [19], [13]) since in our model, the adversary can choose the time instants of corruption while in previous models, the time instants of corruption are chosen randomly through i.i.d. coin flips. Our model of corruption locations are motivated by empirical observations that corruptions rarely occur randomly and are typically bunched [23, 42]. Thus we assume that the adversary has causal power to choose the location of corruptions rather than corruptions occurring at random instants.
Regret as a Performance Measure: The regret of the algorithm denoted by  $\mathrm{Reg}_T$  is defined as
$$
\operatorname {R e g} _ {T} = \sum_ {t = 1} ^ {T} l _ {t} = \sum_ {t = 1} ^ {T} \left\| \theta_ {t} - \theta_ {t} ^ {*} \right\|,
$$
<sup>1</sup>We denote by  $[T] := \{1, \dots, T\}$
2Throughout, we denote by  $\parallel \cdot \parallel$  as the L2 norm operator.
3 Convexity implies existence and uniqueness of  $\theta_t^* \in \Theta$
4In the literature, this is sometimes also denoted as an optimization algorithm. c.f. [8]
the total cumulative loss. This definition is known as clean dynamic regret [13], since  $(i)\theta_{t}$  is compared against the true parameter  $\theta_t^*$ , even if the input  $X_{t}$  at time  $t$  is corrupted, i.e., even if  $C_t\neq 0$ , and  $(ii)\theta_{t}$  is compared to the time-varying optimal  $\theta_t^*$ . Technically the regret of an algorithm  $\mathfrak{A}$  is a random variable that is a function of the probability measures  $(\mathbb{P}_t)^T_{t = 1}$  and the corruption functions  $(\mathcal{C}_t)^T_{t = 1}$  and denoted as  $\mathrm{Reg}_T^{(\mathfrak{A})}((\mathbb{P}_t)^T_{t = 1},(\mathcal{C}_t)^T_{t = 1})$ . We will denote the regret as  $\mathrm{Reg}_T$  whenever it is clear from context.
We now define the data-stream complexity parameters impacting regret:  $(i)$  distribution shift and (ii) corruptions.
Distribution Shift<sup>5</sup>: The cumulative distance between successive optimum points  $\Phi_T := \sum_{t=2}^{T} \|\theta_t^* - \theta_{t-1}^*\|$  measures the complexity of distribution-shift on the data-stream. This is a natural measure since if the estimator outputs  $\theta_t \in \Theta$  at time  $t$ , will incur an  $L2$  loss  $l_t = \|\theta_t - \theta_t^*\|$ .
Corruptions: The corruption measure is the count  $\Lambda_T := \sum_{t=1}^T \mathbf{1}(C_t \neq 0)$ , of the number of times the random sample  $Z_t$  is corrupted before inputting to the estimator. The count  $\Lambda_T$  is a random-variable since the vector  $C_t$  is a deterministic function of past random variables.
With this setup, the main desiderata we seek of an estimator is that it  $(i)$  is online:  $O(d)$  run-time per data point on a stream and  $O(d)$  overall memory,  $(ii)$  is free of distributional knowledge, i.e., does not require bounds on moments of  $(\mathbb{P}_t)_{t\geq 1}$ , distribution shift or corruptions,  $(iv)$  attains sub-linear in  $T$  regret even when the distributions  $(\mathbb{P}_t)_{t\geq 1}$  are heavy-tailed, and  $(v)$  exhibits graceful degradation of regret with respect to distribution shift and corruptions on the data stream.
1. Problem formulation and lower bounds: We formalize the problem of online estimation under drift and corruptions and define a dimension-free target regret benchmark in Equation (1) that captures all the aforementioned desiderata. We give lower bounds in Propositions 16.1 and 2.6 that shows the necessity of assumptions such as finite diameter for the set  $\Theta \subset \mathbb{R}^d$  in which the unknown parameters  $\theta_t^* \in \Theta$  lie. Further discussion in Table 1 and in Section 11.
2. High-probability upper bounds: Theorem 5.1 proves that a tuned version of clipped-SGD achieves all aforementioned desiderata. No previous algorithm achieves all the desiderata simultaneously (Tables 1 and 2). Further in Theorem 4.3, we show that if in addition the data-stream is sub-gaussian, then our bound is optimal in the sense when instantiated with special cases of having no drift and corruptions [24] or in the setting of no corruptions and 0 variance [43], it recovers known optimal results.
3. Algorithmic insights: Our key finding is that neither the learning rate of  $\mathcal{O}(1)$  known to be optimal for SGD to adapt to distribution shift in the absence of noise, nor the  $\mathcal{O}(1 / t)$  learning rate known to be optimal in a stationary environment can be used. Instead a learning rate of  $T^{-\alpha}$  for  $\alpha < 1$  is needed to combat noise and be adaptive to drift simultaneously. See also Figure 1. We show a lower bound in Proposition 6.1 that this insight is fundamental and not an artifact of our analysis. Knowledge of  $T$  is relaxed using the doubling trick in Section 14 in the Appendix. Our result improves prior state of art in [8] that show that in the absence of corruptions, SGD without clipping can give sub-linear regret bounds holding in expectation. However, in the presence of heavy-tailed noise, high probability guarantees are more insightful compared to only in expectation [63]. Our algorithmic contribution is that a tuned version of clipped SGD can give high probability sub-linear regret guarantees, that hold even in the presence of non-stationarities and corruptions.
4. Technical contributions in the proof techniques: From a technical perspective, we employ novel proof techniques to derive high-probability under any learning rate on data-streams exhibiting arbitrary distribution shift. The prior state-of-art analysis in [36, 55] are limited to the specialized setting when the learning rate is  $\eta_t = 1 / (t + 1)$  and the data does not exhibit drift. To be adaptive to drift requires newer techniques even in the case of sub-gaussian tails to achieve dimension-free bounds (cf. Section 4.1). We used an induction based technique to apply martingale concentrations in Lemma 19.9. We build upon this technique using contradiction arguments in Lemma 20.10 to extend to the heavy-tailed case. These arguments are of independent interest to prove bounds for other iterative algorithms.
5Following convention [22], we use the term drift to refer to distribution shift
Setting	Best lower bound	Best upper bound
Stationary, no-corruptions (ΦT=0, ΛT=0)	Ω[√T(√Trace(Σ) +√νmax(Σ) ln(1/δ))] [10]	O[√T(√Trace(Σ) +√νmax(Σ) ln(T/δ))] [38]
Stationary, corruptions (ΦT=0, ΛT≥0)	Prop 2.6: Ω(ΛT D)	Thm 5.1: O(T5/6σ2+T3/4D ΛT)
Non-stationary, no-corruptions (ΦT≥0, ΛT=0).	[8]: Ω(T2/3ΦT1/3)	Thm 5.1: O(T2/3ΦT+T5/6σ2) [8] gives bound in expectation only.
Non-stationary, corruptions (ΦT≥0, ΛT≥0)	Prop 16.1: Ω (ΛT D)	Thm 5.1: O(T2/3ΦT+T5/6σ2+T3/4 ΛT D)
Table 1: Regret bounds for online estimation with heavy-tailed data and strongly convex loss. The setting of  $\Phi_T = 0, \Lambda_T = 0$  (line 1) is characterized (upto log factors) in [10] and [38]. The first dimension free, high probability upper bounds are established in this paper for all other settings. For the setting of  $(\Phi_T \geq 0, \Lambda_T = 0)$ , [8] gives a lower bound and an upper bound holding in expectation, while ours is the first high-probability upper bound. Except line 1, there is a gap between known lower and upper bounds, closing which is an open problem (see also Section 9).
1.3 Motivating application
Online Anomaly Detection (AD) problems consist of a loss function  $\mathcal{L}(\cdot, \cdot)$  and parameter  $\theta_t$  that varies with time  $t$ , such that for the input  $X_t$  received at time  $t$ , the algorithm outputs anomaly score  $S_t \coloneqq \mathcal{L}(X_t, \theta_t)$  ([47], [12]). The objective is to output lower scores for samples that are not corrupted, i.e., if at time  $t$  the data distribution is  $Z_t \sim \mathbb{P}_t$ , then the optimal anomaly score  $S_t^* \coloneqq \mathcal{L}(X_t, \theta_t^*)$  is the one produced by the model minimizing the average anomaly score i.e.,  $\theta_t^* \in \arg \min_{\theta \in \Theta} \mathbb{E}_{Z \sim \mathbb{P}_t}[\mathcal{L}(Z, \theta)]$  [51]. The distance  $\| \theta_t - \theta_t^*\|$  between  $\theta_t$  the model used at time  $t$  and the optimal  $\theta_t^*$  is a measure of AD performance degradation [32, 50], motivating our study of low regret algorithms.
Special cases of such problems include robust online mean estimation [19] and robust online linear regression [20], both of which we consider in simulations in Section 7. The online mean estimation corresponds to  $\mathcal{L}(X,\theta)\coloneqq \frac{1}{2}\| X - \theta \| _2^2$  since arg  $\min_{\theta}\mathbb{E}_{Z\sim \mathbb{P}}[||Z - \theta ||^2 ] = \mathbb{E}_{Z\sim \mathbb{P}}[Z]$ . Online linear regression corresponds to splitting the input  $X\in \mathbb{R}^{d_1 + d_2}$  as  $X\coloneqq (X^{(1)},X^{(2)})$ , with  $X^{(1)}\in \mathbb{R}^{d_1}$  and  $X^{(2)}\in \mathbb{R}^{d_2}$  and using the re-construction loss  $\mathcal{L}(X,\theta)\coloneqq \frac{1}{2}\| X^{(2)} - \theta^T X^{(1)}\| _2^2$  as anomaly score. The linear regression setting of detecting anomalies by reconstructing one half of the input  $X$  from the other is popularly known as self-supervised AD models [52]. Empirically, in [30],  $\mathcal{L}(X,\theta)\coloneqq \| (\mathbb{I}_d - \theta)X\|$  with  $\theta \in \mathbb{R}^{d\times d}$  a matrix of rank  $k < d$  and  $\mathbb{I}_d\in \mathbb{R}^{d\times d}$  the identity matrix is used to detect anomalies. The linear model of [30] is extended to a non-linear setting in [42], using an auto-encoder [65]. In [17], a tree density model for  $\mathcal{L}(X,\theta)$  is studied. These papers' focus is empirical and do not give statistical guarantees.
Organization of the Paper: In Section 2 we state the target benchmark regret along with the assumptions under which we seek it and appropriate lower bounds. In Section 3 we give the clipped-SGD algorithm. In Section 4, we give results under the additional assumption of sub-gaussian tails. This helps build intuition and proof techniques for the general result which we give in Section 5. We state implications of our results in Section 6, provide empirical evidence in Section 7, discuss related work in Section 8 and conclude with open problems in Section 9.
2 Formalizing the desiderata
Before formalizing the desiderata in Equation (1), we state some mild assumptions standard in the study of statistical estimation under which we seek guarantees [55, 19].
2.1 Model Assumptions
Assumption 2.1 (Strong convexity). There exists  $0 < m \leq M < \infty$  known to the algorithm, such that for all  $t$ , the function  $\theta \to \mathbb{E}_{Z \sim \mathbb{P}_t}[\mathcal{L}(Z, \theta)]$  is  $M$  smooth and  $m$  strongly convex.
This is a benign assumption that essentially states that properties of the loss function such as the convexity and smoothness are known to the estimation algorithm.
Methods	Regret	Moment assumed	Drift tolerance	Corruption tolerance	Free of distributional knowledge
[55]	O(σ√T)	2	0	0	x
[36]	O(σT1/p)	p ∈ {1, 2]	0	0	✓
[19]	O(√dT+(dT)1/4 ΛT1/2)	2	0	O(T)	x
Corollary. 4.8 (This paper)	O(T2/3ΦT+T2/3σ +T1/3D1/2 ΛT)	sub-gaussian	O(T1/3)	O(T2/3)	x
Theorem 4.3 (This paper)	O(TαΦT+T2-α/2σ +Tα/2D1/2 ΛT)	sub-gaussian	O(T1-α)	O(T2-α/2)	x6
Corollary 5.3 (This paper)	O(T2/3ΦT+T5/6σ2 +T3/4D1/2 ΛT)	2	O(T1/3)	O(T1/4)	✓
Theorem 5.1 (This paper)	O(T4/3ΦT+T3-α/3σ2 +T3/2D1/2 ΛT)	2	O(T3-4α/3)	O(T2-3α/2)	✓
Table 2: Comparison of high-probability regret bounds of online estimation algorithms of strongly convex functions under various moment assumptions. [55] requires distance of the initial point to the unknown optimum. Theorem 4.3, Corollary 4.8, [19] and [55] requires a bound on  $\sigma$ . Further, [19] is not dimension-free as regret depends on  $d$ . We do not include [8] as their regret bounds only hold in expectation and not with high probability.
Assumption 2.2 (Convex domain). The algorithm knows a closed convex set  $\Theta \subset \mathbb{R}^p$  such that for all  $t$ ,  $\theta_t^* \coloneqq \arg \min_{\theta \in \Theta} \mathbb{E}_{Z \sim \mathbb{P}_t}[\mathcal{L}(Z, \theta)] \in \Theta$  is the true parameter to be estimated at time  $t$ .
Assumption 2.3 (Finite diameter). The diameter  $\mathcal{D} \coloneqq \max_{x, y \in \Theta} \| x - y\|$  of  $\Theta$  is finite, i.e.,  $\mathcal{D} < \infty$ .
This is a necessary assumption when  $\Lambda_T > 0$  (Proposition 2.6). If  $\Lambda_T = 0$ , we relax this assumption in Corollaries 4.5 and 5.4. Let  $\mathcal{R}_t(\theta) = \mathbb{E}_{Z \sim \mathbb{P}_t}[\mathcal{L}(Z, \theta)]$ , denote the population risk at time  $t$ .
Assumption 2.4 (Known finite upper bound on the true gradient). There exists a known finite  $G < \infty$  such that  $\sup_{\theta \in \Theta} \| \nabla \mathcal{R}_t(\theta) \| \leq G$ .
This is necessary for many online optimization algorithms [36, 55, 25, 27].
Assumption 2.5 (Existence of second moment). There exists a matrix  $\Sigma \succeq 0$ , unknown to the algorithm such that for all  $t$  and  $\theta \in \Theta$ , the covariance of the random-vector  $\nabla \mathcal{L}(Z, \theta)$  is bounded by  $\Sigma$ , i.e.,  $\mathbb{E}_{Z \sim \mathbb{P}_t}[(\nabla \mathcal{L}(Z, \theta) - \nabla \mathcal{R}_t(\theta)) \cdot (\nabla \mathcal{L}(Z, \theta) - \nabla \mathcal{R}_t(\theta))^T] \preceq \Sigma$  holds for all  $t$  and  $\theta$ .
The class of distributions admissible by this assumption is large including all sub-gaussian distributions and heavy-tailed distributions that do not have finite higher moments [25], [44]. Observe that no parametric assumptions on the distribution family is made. Throughout, we denote by  $\sigma^2$  and  $\nu_{max}(\Sigma)$  as the trace and the largest eigen-value of  $\Sigma$  respectively. For a positive semi-definite matrix  $\Sigma \in \mathbb{R}^{p\times p}$ ,  $\mathcal{P}(\Sigma)$  is the set of all probability measures satisfying assumptions 2.1, 2.4 and 2.5.
2.2 Target benchmark for algorithm design
We say that an estimator is free of distributional knowledge if it does not need as input bounds on either the moments of  $\mathbb{P}_t$  nor on the stream complexity  $\Phi_T$  and  $\Lambda_T$ . Our key focus - formalized below in Equation (1) - is: can we design an online algorithm without distributional assumptions that has sub-linear regret and that degrades gracefully with drift and corruption?
Can we design an online algorithm  $\mathfrak{A}$  without distributional knowledge, such that  $\exists l,m,n < 1$  such that  $\forall \Sigma \succeq 0\in \mathbb{R}^{p\times p}$ ,  $(\mathbb{P}_t)_t^T = \mathcal{P}(\Sigma)$  and  $(\mathcal{C}_t)_t^T = 1$ , the regret bound
$$
\operatorname {R e g} _ {T} ^ {(\mathfrak {A})} \left(\left(\mathbb {P} _ {t}\right) _ {t = 1} ^ {T}, \left(\mathcal {C} _ {t}\right) _ {t = 1} ^ {T}\right) \leq \mathcal {O} \left(\left(T ^ {l} \Phi_ {T} + T ^ {m} + T ^ {n} \mathcal {D} \Lambda_ {T}\right). \text {p o l y} \left(\sigma , \log \left(\frac {T}{\delta}\right), G, \frac {1}{m}\right)\right) \tag {1}
$$
holds with probability at-least  $1 - \delta$  ?
$^6\alpha \in (0,1]$  is an input parameter.
Throughout, we use  $\mathcal{O}(\cdot)$  to hide poly-logarithmic factors. Our main result is Theorem 5.1, where we prove that tuned clipped SGD achieves this, and our other desiderata.
2.3 Why is Equation (1) a good benchmark for regret bounds with drift and corruption?
The benchmark demands robustness to heavy-tailed data, since regret is sub-linear for any distribution with finite second moment. The benchmark is dimension free, since dimension terms  $d, p$  do not appear in Equation (1). The finite diameter  $\mathcal{D}$  only affects the bound due to corruptions and not the other terms. In particular, if  $\Lambda_T = 0$  almost-surely, then the benchmark seeks a valid bound even if  $\mathcal{D} = \infty$ . In general when  $\Lambda_T > 0$  however, dependence on  $\mathcal{D}$  cannot be avoided as we show in Proposition 2.6 below (proof in Section 16).
Proposition 2.6 (Finite diameter is necessary even in the absence of drift). There exists a strongly convex loss function  $\mathcal{L}$  such that for every  $\mathcal{D} > 0$  and  $d\in \mathbb{N}$ , domain  $\Theta = \{x\in \mathbb{R}^d:\| x\| \leq \mathcal{D}\}$  such that for every  $T\in \mathbb{N}$  and every algorithm, there exists a probability measure  $\mathbb{P}$  on  $\mathbb{R}^d$  from which the un-corrupted samples are drawn from (i.e.,  $\Phi_T = 0$ ) and a sequence of corruption functions  $(\mathcal{C}_t)^T_{t = 1}$  such that the regret is at-least  $\Omega (\Lambda_T\mathcal{D})$  with probability 1.
Thus, even if there is no drift, finite diameter assumption cannot be avoided. Proof is given in Section 16, relies on the modeling assumption that the corruption locations can be arbitrary. This lower bound is in contrast to prior work [19] which shows that finite diameter is not necessary in the absence of corruptions and when the time instants of corruptions are random. Further in Table 1, we compare the best known lower and upper bounds for the various settings in the presence and absence of distribution shifts and corruptions. As can be seen, in all cases involving either distribution shift or corruptions, our work is the first to give high-probability estimation regret bounds.