SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Aaron Sidford and Chenyi Zhang
{sidford, chenyiz}@stanford.edu
We consider the problem of minimizing a continuous function given given access to a natural quantum generalization of a stochastic gradient oracle. We provide two new methods for the special case of minimizing a Lipschitz convex function. Each method obtains a dimension versus accuracy trade-off which is provably unachievable classically and we prove that one method is asymptotically optimal in low-dimensional settings. Additionally, we provide quantum algorithms for computing a critical point of a smooth non-convex function at rates not known to be achievable classically. To obtain these results we build upon the quantum multivariate mean estimation result of Cornelissen et al. [25] and provide a general quantum variance reduction technique of independent interest.
Stochastic optimization is central to modern machine learning. Stochastic gradient descent (SGD), and its many variants, are used broadly for solving challenges in data science and learning theory. In theory, SGD and stochastic optimization, have been the subject of decades of extensive study. [19, 68, 27] established that SGD achieves optimal rates for minimizing Lipschitz convex functions<sup>1</sup> (even in one dimension) and stochastic optimization methods have been established for a range of problems [72, 70, 73, 74]. More recently, the complexity of stochastic gradient methods for smooth non-convex optimization, e.g., critical point computation, were established [3, 29, 30, 49, 50].
Given the foundational nature of stochastic optimization and the potential promise and increased study of quantum algorithms, it is natural to ask whether quantum computation could enable improved rates for solving these problems. There has been work studying whether access to quantum counterparts of classic optimization oracle can yield faster rates for semidefinite programs [13, 12, 43, 75, 76, 54], convex optimization [21, 5, 20], and non-convex optimization [23, 63, 80]. Notably, [52] showed that with just access to the quantum analog of an zeroth-order oracle, i.e., an oracle that when queried at a point outputs the value of the function at that point, it is possible to simulate access to a classic gradient oracle with a single query. This tool immediately yields improved rates for, e.g., convex function minimization, with a zeroth-order oracle.
Unfortunately, despite this progress and the established power of quantum evaluation oracles, obtaining further improvements has been challenging. A line of work established a variety of striking lower bounds ruling out quantum speedups for fundamental optimization problems [31, 32, 82]. For example, [31] showed that when given access to the quantum analog of a first-order oracle, i.e., an oracle that when queried at a point outputs the value of the function as well as the gradient at that point, quantum algorithms have no improved rates for non-smooth convex optimization over GD and SGD when the dimension is large. [82] extended this result to the non-convex setting and
<sup>1</sup>We assume all objective functions are differentiable. Similar to related work, see e.g., [15], our results can be generalized to non-differentiable settings since convex functions are almost everywhere differentiable and our algorithms and corresponding convergence analysis are robust to polynomially small numerical errors.
showed that given access to the quantum analog of a stochastic gradient oracle, quantum algorithms have no improved rates for finding critical points over SGD when the dimension is large.
In spite of these negative results, we nevertheless ask, for stochastic optimization are quantum-speedups obtainable? Our main result is an answer to this question in the affirmative for dimension-dependent algorithms. We provide two different quantum algorithms for stochastic convex optimization (SCO) which provably outperform optimal classic algorithms. Furthermore, we provide a quantum algorithm for computing the critical point of a smooth non-convex function which improves upon the state-of-the-art. We obtain these results through a new general quantum-variance reduction technique built upon the quantum multivariate mean estimation result of Cornelissen et al. [25] and the multilevel Monte Carlo (MLMC) technique [41, 10, 7]. We complement these results with lower bounds showing that one of them is asymptotically optimal in low-dimensional settings.
General notation. We use  $\| \cdot \|$  to denote the Euclidean norm and let  $\mathbb{B}_R(\mathbf{x})\coloneqq \{\mathbf{y}\in \mathbb{R}^d:\| \mathbf{y} - \mathbf{x}\| \leq R\}$  and  $[T]\coloneqq \{1,\ldots ,T\}$ . We use bold letters, e.g.,  $\mathbf{x},\mathbf{y}$ , to denote vectors and capital letters, e.g.,  $A,B$ , to denote matrices. For a  $d$ -dimensional random variable  $X$ , we refer to the trace of the covariance matrix of  $X$  as its variance, denoted by  $\operatorname{Var}[X]$ . For  $f\colon \mathbb{R}^d\to \mathbb{R}$ , we let  $f^{*}\coloneqq \inf_{\mathbf{x}}f(\mathbf{x})$  and call  $\mathbf{x}\in \mathbb{R}^d$ $\epsilon$ -(sub)optimal if  $f(\mathbf{x})\leq f^{*} + \epsilon$  and  $\epsilon$ -critical if  $\| \nabla f(\mathbf{x})\| \leq \epsilon$ . Moreover, we call a random point  $\mathbf{x}\in \mathbb{R}^d$  expected  $\epsilon$ -(sub)optimal if  $\mathbb{E}f(\mathbf{x})\leq f^{*} + \epsilon$  and expected  $\epsilon$ -critical if  $\mathbb{E}\| \nabla f(\mathbf{x})\| \leq \epsilon$ .  $f\colon \mathbb{R}^d\to \mathbb{R}$  is  $L$ -Lipschitz if  $f(\mathbf{x}) - f(\mathbf{y})\leq L\| \mathbf{x} - \mathbf{y}\|$  for all  $\mathbf{x},\mathbf{y}\in \mathbb{R}^d$  and  $\ell$ -smooth if  $\| \nabla f(\mathbf{x}) - \nabla f(\mathbf{y})\| \leq \ell \| \mathbf{x} - \mathbf{y}\|$  for all  $\mathbf{x},\mathbf{y}\in \mathbb{R}^d$ . For two matrices  $A,B\in \mathbb{R}^{d\times d}$ ,  $A\spreceq B$  denotes that  $\mathbf{x}^T A\mathbf{x}\leq \mathbf{x}^T B\mathbf{x}$  for all  $\mathbf{x}\in \mathbb{R}^d$ . We use  $\tilde{\mathcal{O}}$  to denote the big- $\mathcal{O}$  notation omitting poly-logarithmic factors in  $\epsilon, \epsilon', d, \sigma, \hat{\sigma}, R,$  and  $L$ . When applicable, we use  $|\mathrm{garbage}(\cdot))$  to denote possible garbage states² that arise during the implementation of a quantum oracle (e.g., in Definitions (1) and (2)).
1.1 Quantum stochastic optimization oracles
Here we formally define quantum stochastic optimization oracles that we study in this work.
Qubit notation. We use  $|\cdot \rangle$  to represent input or output registers made of qubits that could be in superpositions. In particular, given  $m$  points  $\mathbf{x}_1,\ldots ,\mathbf{x}_m\in \mathbb{R}^d$  and a coefficient vector  $\mathbf{c}\in \mathbb{C}^m$  with  $\sum_{i\in [m]}|c_i|^2 = 1$ , the quantum register could be in the quantum state  $|\psi \rangle = \sum_{i\in [m]}c_i|\mathbf{x}_i\rangle$ , which is superposition over all these  $m$  points at the same time. If we measure this state, we will get each  $\mathbf{x}_i$  with probability  $|c_i|^2$ . Furthermore, to model a classical probability distribution  $p$  over  $\mathbb{R}^d$  quantumly, we can prepare the quantum state  $\int_{\mathbf{x}\in \mathbb{R}^d}\sqrt{p(\mathbf{x})\mathrm{d}\mathbf{x}} |\mathbf{x}\rangle$ . If we measure this state, the measurement outcome would follow the probability density function  $p$ .
Quantum random variable access. We say that we have quantum access to a  $d$ -dimensional random variable  $X$  if we can query the following quantum sampling oracle of  $X$  that returns a quantum superposition over the probability distribution of  $X$  defined as follows. $^3$
Definition 1 (Quantum sampling oracle). For a  $d$ -dimensional random variable  $X$ , its quantum sampling oracle  $O_X$  is defined as
$$
O _ {X} | 0 \rangle \rightarrow \int_ {\mathbf {x} \in \mathbb {R} ^ {d}} \sqrt {p _ {X} (\mathbf {x}) \mathrm {d} \mathbf {x}} | \mathbf {x} \rangle \otimes | \text {g a r b a g e} (\mathbf {x}) \rangle , \tag {1}
$$
where  $p_X(\cdot)$  represents the probability density function of  $X$ .
The garbage state in Definition 1 is a quantum analogue of classical garbage information that arises when preparing the classical sampling oracle of  $X$ . When implementing the quantum sampling
2The garbage state is a quantum analogue of classical garbage information that arises when preparing the classical stochastic gradient oracle which cannot be erased or uncomputed in general. In this work, we consider a general model where we make no assumption on the garbage state. See e.g., [42] for a similar discussion of this standard use of garbage quantum states.
3 Throughout this paper, whenever we have access to a quantum oracle  $O$ , we assume that it is a unitary operation and that we also have access to its corresponding inverse operation, denoted as  $O^{-1}$ , that satisfies  $O^{-1}O = OO^{-1} = I$ . These are a standard assumption, either explicitly or implicitly, in prior work on quantum algorithms, see e.g., [14, 44, 25].
oracle in quantum superpositions however, this garbage information will appear in a quantum state and cannot be erased or uncomputed in general. In this work, we consider a general model where we make no assumption on the garbage state. See e.g., [42] for a similar discussion of this standard use of garbage quantum states.
Observe that if we directly measure the output of  $O_{X}$ , it will collapse to a classical sampling access to  $X$  that returns random vectors with respect to the probability distribution  $p_{X}$ .
Quantum stochastic gradient oracle. When considering the problem of optimizing a function  $f\colon \mathbb{R}^d\to \mathbb{R}$ , we are often given access to a stochastic gradient oracle that returns a random vector from the probability distribution of the stochastic gradient.
Definition 2 (Stochastic gradient oracle (SGO)). For  $f\colon \mathbb{R}^d\to \mathbb{R}$ , its stochastic gradient oracle (SGO)  $\mathcal{C}_{\mathbf{g}}$  is defined as a random function that when queried at  $\mathbf{x}$ , samples a vector  $\mathbf{g}(\mathbf{x})$  from a probability distribution  $p_{f,\mathbf{x}}(\cdot)$  over  $\mathbb{R}^d$  that satisfies
$$
\underset {\mathbf {g} (\mathbf {x}) \sim p _ {f, \mathbf {x}}} {\mathbb {E}} \mathbf {g} (\mathbf {x}) = \nabla f (\mathbf {x}), \quad \forall \mathbf {x} \in \mathbb {R} ^ {d}.
$$
We say the oracle is  $L$ -bounded if
$$
\underset {\mathbf {g} (\mathbf {x}) \sim p _ {f, \mathbf {x}}} {\mathbb {E}} \| \mathbf {g} (\mathbf {x}) \| ^ {2} \leq L ^ {2}, \quad \forall \mathbf {x} \in \mathbb {R} ^ {d},
$$
and we say the oracle has variance  $\sigma^2$  if
$$
\underset {\mathbf {g} (\mathbf {x}) \sim p _ {f, \mathbf {x}}} {\mathbb {E}} \| \mathbf {g} (\mathbf {x}) - \nabla f (\mathbf {x}) \| ^ {2} \leq \sigma^ {2}, \quad \forall \mathbf {x} \in \mathbb {R} ^ {d}.
$$
In this paper, we further assume quantum access to a stochastic gradient oracle, or access to a quantum stochastic gradient oracle for brevity, that upon query returns a quantum superposition over the probability distribution  $p_{f,\mathbf{x}}(\mathbf{v})$ .
Definition 3 (Quantum stochastic gradient oracle (QSGO)). For  $f\colon \mathbb{R}^d\to \mathbb{R}$ , its quantum stochastic gradient oracle (QSGO) is defined as
$$
O _ {\mathbf {g}} | \mathbf {x} \rangle \otimes | 0 \rangle \rightarrow | \mathbf {x} \rangle \otimes \int_ {\mathbf {v} \in \mathbb {R} ^ {d}} \sqrt {p _ {f , \mathbf {x}} (\mathbf {v}) \mathrm {d} \mathbf {v}} | \mathbf {v} \rangle \otimes | \text {g a r b a g e} (\mathbf {v}) \rangle , \tag {2}
$$
where  $p_{f,\mathbf{x}}(\cdot)$  is as defined in Definition 2.
Observe that if we directly measure the output of  $O_{\mathbf{g}}$ , it will collapse to a classical stochastic gradient oracle that randomly returns a stochastic gradient at  $\mathbf{x}$ .
Another standard assumption in previous works [1, 29, 30, 50] that the stochastic gradient can be queried simultaneously, which means that the algorithm can choose the random seed  $\omega$  that is being queried. We assume that in such setting there is an explicit probability distribution  $\omega$  such that  $\mathbb{E}[\mathbf{g}(\mathbf{x},\omega)] = \nabla f(\mathbf{x})$ . Similarly, we define the quantum access to stochastic gradients allowing simultaneous queries, or access to a quantum stochastic gradient oracle with simultaneous queries for brevity, that upon query returns  $\mathbf{g}(\mathbf{x},\omega)$  in a quantum state.
Definition 4 ( $\sigma$ -SQ-QSGO). For  $f \colon \mathbb{R}^d \to \mathbb{R}$  with its stochastic gradient  $\mathbf{g}(\mathbf{x}, \omega)$  indexed by random seed  $\omega$  that satisfies
$$
\underset {\boldsymbol {\omega}} {\mathbb {E}} \mathbf {g} (\mathbf {x}, \omega) = \nabla f (\mathbf {x}), \underset {\boldsymbol {\omega}} {\mathbb {E}} \| \mathbf {g} (\mathbf {x}, \omega) - \nabla f (\mathbf {x}) \| ^ {2} \leq \sigma^ {2}, \forall \mathbf {x} \in \mathbb {R} ^ {d}, a n d
$$
$$
\underset {\omega} {\mathbb {E}} \| \mathbf {g} (\mathbf {x}, \omega) - \mathbf {g} (\mathbf {y}, \omega) \| ^ {2} \leq \ell^ {2} \| \mathbf {x} - \mathbf {y} \| ^ {2}, \quad \forall \mathbf {x}, \mathbf {y} \in \mathbb {R} ^ {d} \tag {3}
$$
its simultaneously queriable,  $\sigma$ -mean-squared smooth quantum stochastic gradient oracle ( $\sigma$ -SQ-QSGO) is defined as
$$
O _ {\mathbf {g}} ^ {S} | \mathbf {x} \rangle \otimes | \omega \rangle \otimes | 0 \rangle \rightarrow | \mathbf {x} \rangle \otimes | \omega \rangle \otimes | \mathbf {g} (\mathbf {x}, \omega) \rangle \otimes | \operatorname {g a r b a g e} (\mathbf {x}, \omega) \rangle . \tag {4}
$$
1.2 Results
Here we present our main results on new quantum algorithms for stochastic optimization. Our results and the prior state-of-the-art are summarized in Table 1. Further, we discuss new quantum lower bounds that we establish for quantum variance reduction and stochastic convex optimization. Our algorithmic results leverage a common technique for quantum variance reduction introduced in the next Section 2. This technique uses a combination of the quantum multivariate mean estimation result of Cornelissen et al. [25] and multilevel Monte Carlo (MLMC) [41, 10, 7].
Setting	Queries	Output	Method
Convex	ε-2	ε-optimal	SGD
Convex	d5/8ε-3/2	ε-optimal	Our Result (Q-AC-SA, Algorithm 2)
Convex	d3/2ε-1	ε-optimal	Our Result (Q-SCP, Section 4)
Non-convex (Bounded variance)	ε-4	ε-critical	Randomized SGD [40]
Non-convex (Bounded variance)	d1/2ε-3	ε-critical	Our Result (Q-SGD, Algorithm 6)
Non-convex (Mean-squared smoothness)	ε-3	ε-critical	SPIDER [29]
Non-convex (Mean-squared smoothness)	d1/2ε-5/2	ε-critical	Our Result (Q-SPIDER, Algorithm 7)
Table 1: Comparison between our quantum algorithms and state-of-the-art classical algorithms for Problem 1 (the convex setting) and Problem 2 (the non-convex setting) where, for simplicity, we assume  $R = L = 1$  for Problem 1 and  $\ell = \sigma = \delta = 1$  for Problem 2. The  $\widetilde{\mathcal{O}}$  symbol was omitted in the "Queries" column.
Stochastic convex optimization. In this work we consider the quantum analog of the standard stochastic convex optimization (SCO) problem defined as follows.
Problem 1 (Quantum stochastic convex optimization (QSCO)). In the quantum stochastic convex optimization (QSCO) problem we are given query access to an  $L$ -bounded QSGO  $O_{\mathbf{g}}$  (see Definition 3) for a convex function  $f \colon \mathbb{R}^d \to \mathbb{R}$  whose minimum is achieved at  $\mathbf{x}^*$  with  $\| \mathbf{x}^* \| \leq R$  and must output an expected  $\epsilon$ -optimal point.
Classically, it is known that simple stochastic gradient descent, e.g.  $x_{t + 1} = x_t - \eta g_t$ , can solve QSCO with  $\mathcal{O}(\epsilon^{-2})$  queries. Further, this bound is known to be optimal in the worst case [69, 77].
Nevertheless, we develop two quantum algorithms for Problem 1 in Section 3 and Section 4, respectively. The query complexities of these algorithms are summarized in the following Theorem 1. In comparison to the optimal classical query complexity of  $\mathcal{O}(\epsilon^{-2})$  queries, our algorithms obtain an improved dependence in terms of  $\epsilon$  at the cost of a worse dependence on the dimension,  $d$ ; Theorem 1 shows that a quadratic speedup is achievable when  $d$  is constant.
Theorem 1 (Informal version of Theorem 5 and Corollary 1). Problem 1 can be solved using an expected  $\widetilde{\mathcal{O}} (\min \{d^{5 / 8}(LR / \epsilon)^{3 / 2},d^{3 / 2}LR / \epsilon \})$  queries.
We complement Theorem 1 with the following lower bound on the query complexity for QSCO.
Theorem 2 (Informal version of Theorem 8). For any  $\epsilon \leq \mathcal{O}(d^{-1/2})$ , any quantum algorithm that solve Problem 1 with probability at least  $2/3$  makes at least  $\Omega(d^{1/2}\epsilon^{-1})$  queries in the worst case. For any  $\Omega(d^{-1/2}) \leq \epsilon \leq 1$ , any quantum algorithm that solves Problem 1 with success probability at least  $2/3$  must make at least  $\Omega(\epsilon^{-2})$  queries in the worst case.
Theorem 2 shows that the  $\widetilde{\mathcal{O}}(d^{3/2} LR/\epsilon)$  rate that we obtain is asymptotically optimal for  $d = \widetilde{\mathcal{O}}(1)$ . A key open problem is whether the dimension dependence in either our upper bounds (Theorem 5 and Corollary 1) or our lower bound (Theorem 8) can be improved.
Stochastic critical point computation. We also develop quantum algorithms for finding critical points, i.e., points with small gradients, of (possibly) non-convex functions.
Problem 2 (Quantum stochastic critical point computation (QSCP)). In the quantum stochastic critical point computation (QSCP) problem, for an  $\ell$ -smooth (possibly) non-convex  $f\colon \mathbb{R}^d\to \mathbb{R}$  satisfying  $f(\mathbf{0}) - \inf_{\mathbf{x}}f(\mathbf{x})\leq \Delta$  we are given query access to one of the following two oracles
1. (Bounded variance setting). A QSGO  $O_{\mathbf{g}}$  with variance  $\sigma^2$  (see Definition 3), or
2. (Mean-squared smoothness setting). A  $\sigma$ -SQ-QSGO  $O_{\mathbf{g}}^{S}$  (see Definition 4),
and must output an expected  $\epsilon$ -critical point.
Leveraging [40] and [29] we develop two quantum algorithms that solve Problem 2 in the bounded variance setting and the mean-squared smoothness setting, and obtain the following result.
Theorem 3 (Informal version of Theorem 6 and Theorem 7). In the bounded variance setting, Problem 2 can be solved using an expected  $\widetilde{\mathcal{O}}\left(\Delta \ell \sigma d^{1/2} \epsilon^{-3}\right)$  queries. In the mean-squared smoothness setting, Problem 2 can be solved using an expected  $\widetilde{\mathcal{O}}\left(\ell \Delta(d\sigma)^{1/2} \epsilon^{-5/2}\right)$  queries.
In the bounded variance setting, Problem 2 can be solved using  $\mathcal{O}(\epsilon^{-4})$  queries to a classical SGO with variance  $\sigma^2$  (Definition 2) [40], which is known to be optimal [6]. In comparison, our algorithm improves in terms of  $\epsilon$  and achieves a quantum speedup when  $d \leq \mathcal{O}(\epsilon^{-2})$ . In the mean-squared smoothness setting, Problem 2 can be solved using  $\mathcal{O}(\epsilon^{-3})$  queries to a classical stochastic gradient oracle with variance  $\sigma^2$  that satisfies if it satisfies (3) (Definition 2) [29], which is known to be optimal [6]. In comparison, our algorithm improves in terms of  $\epsilon$  and achieves a quantum speedup when  $d \leq \mathcal{O}(\epsilon^{-1})$ .
Quantum zeroth-order oracles. Throughout the paper, we focus on stochastic gradient oracles in correspondence with classical work on stochastic optimization. However, it is worth noting that in certain cases our results extend gracefully to quantum stochastic zeroth order oracles. For example, when the objective function exhibits a finite-sum structure and we have access each component function individually through a quantum zeroth-order oracle, we can achieve an SQ-QSGO (Definition 4) with just a single query, utilizing quantum gradient estimation [52]. However, in other cases, the correspondence is less clear. For instance, if we are given a quantum stochastic zeroth-order oracle where the function value is obfuscated by some external noise, quantum gradient estimation [52] is not directly applicable. Further study could be an interesting direction for future work.
Practicality. Regarding the utility of our algorithm in practical situations, note that our quantum oracles in Definition 3 and Definition 4 are defined as direct, natural generalizations of the corresponding classical oracles. Considering such quantum generalizations of classical oracles is standard in the literature, see e.g., [21, 81]. There are standard techniques for implementing such quantum analogs of classical oracles (in theory for now given the current state-of-the-art in implementing quantum algorithms in practice). In particular, if there is a classical circuit for the classical oracle, there is a standard technique to obtain a quantum circuit of the same size which implement the corresponding quantum oracle and its inverse. Hence, our quantum algorithms have the potential to surpass blackbox classical algorithms in low dimensional settings where the oracle is given as an explicit circuit.
Dimension dependence. Regarding potential concerns regarding the dependence of our quantum algorithms on the problem dimension, below we provide several supplementary points of context.
- As discussed, in the classical setting, prior research [27] demonstrated that when optimizing an 1-Lipschitz convex function, SGD has a query complexity of  $\mathcal{O}(\epsilon^{-2})$  which is optimal, even in the one-dimensional case. In the quantum setting, we showed that, theoretically, quantum speedups which offer a different tradeoff between  $\epsilon$  and  $d$  dependencies are possible. Additionally, from our lower bound presented in Theorem 2 we know that some dimension dependence is inherent in obtaining an improvement.
- Classically, the complexity of dimension dependent optimization methods is well studied. In particular, there are parallel and private stochastic convex settings where the dimension dependencies are discussed, see e.g., [15, 18], and there exists works on critical point computation in low dimension settings, see e.g., [22].
- Even for high-dimensional problems, our algorithms can potentially be used as subroutines for low-dimensional subproblems. For instance, in this paper we apply these method to the approximately best point problem (Problem 5), wherein we utilize our algorithm repeatedly within a one-dimensional setting.
1.3 Paper organization
In the next Section 2 we propose and develop a new algorithm for a new problem quantum variance reduction; this algorithm is the basis of our quantum speedups for stochastic optimization problems. We then discuss the application of quantum variance reduction for stochastic convex optimization by presenting two quantum algorithms in Section 3 and Section 4, respectively. Technically, it is possible to obtain our result in Section 4 just using the quantum mean estimation routine of [25] rather than quantum variance reduction, however our use of quantum variance reduction facilitates our presentation. Furthermore, we present quantum algorithms for non-convex optimization based on quantum variance reduction in Section 5. Finally, in Section 6 we prove quantum lower bounds for quantum variance reduction and quantum stochastic convex optimization that establish the optimality of our algorithms, and conclude the paper in Section 7.
2 Quantum variance reduction
We obtain our quantum speedups for stochastic optimization problems by proposing and developing new algorithms for a new problem which we call the quantum variance reduction problem.
Problem 3 (Variance reduction). For a  $d$ -dimensional random variable  $X$  with  $\operatorname{Var}[X] \leq L^2$  and some  $\hat{\sigma} \geq 0$ , suppose we are given access to its quantum sampling oracle  $O_X$  defined in Definition 1. The goal is to output an unbiased estimate  $\hat{\mu}$  of  $\mu := \mathbb{E}[X]$  satisfying  $\mathbb{E}\|\hat{\mu} - \mu\|^2 \leq \hat{\sigma}^2$ .